r 1 ^ r ^tatc Collesc of jagriculturc ^t Cornell ©nibersiti* aitftaca. B. S. Xi6rarj> Date Due <. oz-o Cornell University Library QA 308.M98 An elementary course in the integral cal 3 1924 002 947 418 LIBRARY OCT 16 1937 AGRIC. f-CON. A, FARM iv^GT. The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924002947418 THE CORNELL MATHEMATICAL SERIES LUCIEN AUGUSTUS WAIT • ■ • Gkxeual Editor (senior professor of mathematics in CORNELL UNIVERSITY) The Cornell Mathematical Series, lucien augustus wait, (Senior Professor of Mathematics in Cornell Vmiiersity,) GENERAL EDITOR. This series is designed primarily to meet the needs of students in En- gineering and Architecture in Cornell University ; and accordingly many practical problems in illustration of the fundamental principles play an early and important part in each book. While it has been the aim to present each subject in a simple manner, yet rigor of treatment has been regarded as more important than simplicity, and thus it is hoped that the series will be acceptable also to general students of Mathematics. The general plan and many of the details of each book were discussed at meetings of the mathematical staff. A mimeographed edition of each vol- ume was used for a term as the text-book in all classes, and the suggestions thus brought out were fully considered before the work was sent to press. The series includes the following works : ANALYTIC GEOMETRY. By J. H. Tanner and Joseph Allen. DIFFERENTIAL CALCULUS. By James McMahon and Virgil Snyder. INTEGRAL CALCULUS. By D. A. Murray. AN ELEMENTARY COURSE INTEGRAL CALCULUS BT DANIEL ALEXANDER MURRAY, Ph.D. INSTKUCTOR IN MATHEMATICS IN CORNELL UNIVERSITY FORMERLY SCHOLAR AND FELLOW OF JOHNS HOPKINS UNIVERSITY AUTHOR OF " INTRODUCTORY COURSE IN DIFFERENTIAL EQUATIONS " oXKo NEW YORK-:. CINCINNATI:- CHICAGO AMERICAN BOOK COMPANY Copyright, 1898, by AMEIUCAN BOOK COMPANY. murkay's integ. calc. w. p. 4 (^ / ^ -f y^-C PREFACE This book has been written primarily for use at Cornell University and similar institutions. In this university the classes in calculus are composed mainly of students in engi- neering for -whom an elementary course in the Integral Calculus is prescribed for the third term of the first year. Their pur- pose in taking the course is to acquire facility in performing easy integrations and the power of making the simple applica- tions which arise in practical work. While the requirements of this special class of students have been kept in mind, care has also been taken to make the book suitable for any one be- ginning the study of this branch of mathematics. The volume contains little more than can be mastered by a student of average ability in a few months, and an effort has been made to present the subject-matter, which is of an elementary char- acter, in a simple manner. The object of the first two chapters is to give the student a clear idea of what the Integral Calculus is, and of the uses to which it may be applied. As this introduction is somewhat longer than is usual in elementary works on the calculus, some teachers may, perhaps, prefer to postpone the reading of sev- eral of the articles until the student has had a certain amount of practice in the processes of integration. It is believed, how- ever, that a careful study of Chapters I., II., will arouse the stu- dent's interest and quicken his understanding of the subject. There may be some difference of opinion also as to whether VI PREFACE the beginner should be introduced to the subject through Chapter I. or through Chapter II. The decision of this ques- tion will depend upon the point of view of the individual teacher. So far as the remaining portion of the book is concerned it is a matter of indifference which of these chapters is taken first; and, with slight modifications, they can be interchanged. In Chapter III. the fundamental rules and methods of integration- are explained. Since it has been deemed advisable to intro- duce practical applications as early as possible. Chapter IV. is devoted to the determination of plane areas and of volumes of solids of revolution. The subject of Integral Curves, which is of especial importance to the engineer, is treated in Chapter XII. Many of the exampleg are original. Others, especially some of those given in the practical applications, by reason of their nature and importance, are common to all elementary courses on calculus. In several instances, examples of particular interest have been drawn from other works. A list of lessons suggested for a short course of eleven or twelve weeks is given on page viii. This list has been arranged so that four lessons and a review will be a week's work. It is hardly possible to name all the sources from which the writer of an elementary work may have obtained suggestions and ideas. I am especially conscious, however, of my indebted- ness to the treatises of De Morgan, Williamson, Edwards, Stegemann and Kiepert, and Lamb. To my colleagues in the department of mathematics at Cornell University, I am under obligations for many valuable criticisms and suggestions. Both the arrangement and the contents have been influenced in a large measure by our con- ferences and discussions. As originally projected, the volume was to have been written in collaboration with Dr. Hutchinson, but circumstances prevented the carrying-out of this plan. PREFACE vii Chapters V., VI., in part, and Articles 28, 73, in their entirety, have been contributed by him. My colleagues have aided me also in correcting the proofs. From Professor I. P. Church of the College of Civil Engi- neering and Professor W. P. Durand of the Sibley College of Mechanical Engineering, I have received valuable suggestions for making the book useful to engineering students. Pro- fessor Durand kindly placed at my disposal, with other notes, his article on " Integral Curves " in the Sibley Journal of Engineering, Vol. XI., No. 4 ; and Chapter XII. is, with slight changes, a reproduction of that article. I take this oppor- tunity of thanking Mr. A. T. Bruegel, Instructor in the kine- matics of machinery, and Mr. Murray Macneill, Eellow in mathematics in this university, the former for the interest and care taken by him in drawing the figures, the latter for his assistance in verifying examples and reading proof sheets. D. A. MURRAY. Cornell Univeksitt. LIST OF LESSONS SUGGESTED FOR A SHORT COURSE PlEVIEWS TO FOLLOW EVERY FOURTH LeSSOn] 1. Arts. 1, 2, 3. 2. Arts. 4, 5, 6, 7. 3. Arts. 8, 9, 10, 11. 4. Arts. 12, 17, 18, to Ex. 9. 5. Arts. 13, 18, Exs. 10-12, 19. 6. Arts. 14, 20. 7. Arts. 15, 16, 21. 8. Arts. 22, 23, Exs. 1-23, odd examples. 0. Art. 23, Exs. 24-32, 24, 25, Exs. 1-8, page 55. 10. Pages 56, 57, even or odd exam- ples, Exs. 9-41, Exs. 42-47. 11. Arts. 26, 27, Exs. 1, 2, page 68. 12. Arts. 28, 29, Exs. 3, 4, page 68. 13. Exs. 5-14, page 68. 14. Art. 30. 15. Arts. 31, 32. 16. Page 76, Exs. 1-15. 17. Pages 76, 77, Exs. 16-26. 18. Arts. 33, 34. 19. Arts. 35, 36. Selected examples. 20. Arts. 37, 38, 39, 40. 21. Arts. 42, 43. 22. Art. 45. 23. Selected examples, pages 98, 99. 24. Arts. 46, 47, 48 (a). 25. Arts. 48 (rf), 49, 50 (a), (c). 26. Art. 51 (a). 27. Arts. 52, 53. 28. Arts. 58, 59, 60. 29. Arts. 61, 02. 30. Arts. 64, 85, 66. 31. Arts. 67, 68, 69. 32. Arts. 63, 70. Selected exam- ples, pages 164, 165. 33. Art. 71. Selected examples, page 165. 34. Art. 72. Selected examples, page 166. 35. Art. 73. 36. Art." 74. Selected examples, pages 164, 165. 37. Art. 75, Ex. 9, page 164. 38. Art. 76. Selected examples. Art. 77, Ex. 1. 39. Selected examples, pages 162- 166. 40. Arts. 78, 79. Selected examples. 41. Art. 80. Selected examples. 42. Arts. 81, 82, 84, 85. 43. Arts. 86, 87, 88. 44. Arts. 89, 91, 92, 94, 95. CONTENTS CHAPTER I Integration a Process of Summation Art. Page 1. Uses of the integral calculus. Definition and sign of integration 2. Illustrations of the summation of infinitesimals 3. Geometrical principle .... 4. Fundamental theorem. Definite integral 5. Supplement to Art. 3 6. Geometrical representation of an integral 7. Properties of definite integrals CHAPTER II Integration the Inverse of Differentiation 8. Integration the inverse of differentiation ..... 18 9. Indefinite integral. Constant of integration 22 10. Geometrical meaning of the arbitrary constant of integration . 23 11. Relation between the indefinite and the definite integral . 24 12. Examples that involve anti-differentials . ... 25 13. iVnother derivation of the integration formula for an area . . 27 14. A new meaning of y in tlie curve whose equation is y =/(x). De- rived curves 29 15. Integral curves 33 16. Summary 35 CHAPTER III Fundamental Rules xnd Methods of Integration 18. Fundamental integrals 36 19. Two universal formulae of integration 39 20. Integration aided by a change of the independent variable . . 41 21. Integration by parts 44 ■1'2. Additional standard forms 47 23. Derivation of the additional standard forms 48 iz X CONTENTS Art. Page 24. Integration of a total differential 52 25. Summary 54 Examples on Chapter III 55 CHAPTER IV Geometrical Applications of the Calculus 26. Applications of the calculus ..... 27. Areas of curves, rectangular coordinates 28. Precautions to be taken in finding areas by integration 29. Precautions to be taken in evaluating definite integrals 30. Volumes of solids of revolution .... 31. On the graphical representation of a definite integral 32. Derivation of the equations of certain curves . Examples on Chapter IV. 68 58 63 67 69 73 74 76 CHAPTER V Rational Fractions 34. Case 1 7',i 36. Case II 81 36. Case III 82 CHAPTER VI Irrational Functions 38. The reciprocal substitution 84 39. Trigonometric substitutions 85 40. Expressions containing fractional powers oi a + hx only . . 86 41. Functions of the form / 1 x^, (« + ftx^) " | . x (fe, in which m, n, are integers 87 42. Functions of the form F(x, Vx^ + ax + b) dx, F{u, v) being a rational function of u, v . . . . . . .88 43. Functions of the form /(x, V-x^ + ax + 6) dx, f{u, v) being a rational function of m, «... . . .89 44. Particular functions involving Vax^ + bx + c . . . 91 45. Integration of x"'(a + bx'^ydx : (a) by the method of undetermined coefficients ; (b) by means of reduction formulae . . .93 Examples on Chapter VI 98 CONTENTS XI 47. 48. 49. 60. 51. 52. 53. 54. 55. 50. 57. CHAPTER VII Integkation or Trigonometric and Exponent I ahvxdx, I cos" zdx, n being an integer Algebraic transformations isecxdxAcosec^xdx it&wxdxAcoV'xdx .... sin" X cos" a; da; lAL Functions J Integration of sin™ a; cos" fte: (a) by tlie metliod of undetermined coefficients ; (6) by means of reduction formulae I tan»»xsec"xcia;, | cot'"xcosec"a:(Jx Use of multiple angles dx J" I; a'cos^x + fi^sin^x dx C dx + 6 cos X 1 e"' sin nxdx, J a - + 6sins ;, \ e"" cos nxdx sin mx cos nx dx, I cos mx cos nx dx,. sinmxsinmxdx CHAPTER VIII Successive Integration. Multiple Integrals 58. Successive integration . 59. Successive integration with respect to a single independent variable 60. Successive integration with respect to two or more independent variables . 61. Application of successive integration to the measurement of areas : rectangular ooiSrdinates 62. Application of successive integration to the measurement of volumes : rectangular coordinates 63. Further application of successive integration to the measurement of volumes : polar coordinates CHAPTER IX Further Geometrical Applications. Mean Values 65. Derivation of the equations of curves in polar coordinates 66. Areas of curves when polar coordinates are used : by single inte- gration Page 100 103 103 106 107 109 113 114 115 115 117 118 119 119 123 126 128 131 134 135 142 xn CONTENTS Akt. Pash 67. Areas of curves when polar coordinates are used : by double integration 1^" 68. Areas in Cartesian coordinates with oblique axes . . • 140 69. Integration after a change of variable. Integration when the vari- ables are expressed in terms of another variable . . . 141 70. Measurement of the volumes of solids by means of infinitely thin cross-sections 71. Lengths of curves : rectangular coordinates 144 72. Lengths of curves : polar coordinates 147 73. The intrinsic equation of a curve .... . ■ 149 74. Areas of surfaces of solids of revolution . .... 152 75. Areas of surfaces whose equations have the form z=f(x,y) . 156 76. Mean values . . 160 77. A more general definition of mean value 163 Examples on Chapter IX 164 CHAPTER X Applications to Mechanics 78. Mass and density 167 79. Center of mass 168 80. Moment of inertia. Radius of gyration 173 CHAPTER XI Approximate Integration. Integration by Mhans of Series. Integra- tion BY Means of the Measurkment of Areas 81. Approximate integration ... 177 82. Integration in series 177 83. Expansion of functions by means of integration in series . . 179 84. Evaluation of definite integrals by the measurement of areas . 181 85. The trapezoidal rule 182 86. Tlie parabolic, or Simpson's one-third rule 184 87. Durand's rule 187 88. The planimeter 188 CHAPTER XII Integral Curves 89. Introduction . 190 90. Special case of differentiation under the sign of integration . . 190 91. Integral curves defined. Their analytical relations . . . 192 92. Simple geometrical relations of integral curves .... 194 CONTENTS xiii Art. Page 93. Simple mechanical relations and applications of integral curves. Successive moments of an area about a line .... 195 94. Practical determination of an integral curve from its fundamental curve. The integraph 198 95. The determination of scales 200 CHAPTER XIII OkDINARY DlFFKRENXrAL EQUATIONS 96. DiSerential equation. Order. Degree 201 97. Constants of integration. General and particular solutions. Deri- vation of a differential equation 202 Section I. Equations of the First Order and the First Degree 98. Equations in which the variables are easily separable . . . 205 99. Equations homogeneous in x and y 205 100. Exact differential equations 206 101. Equations made exact by means of integrating factors . . . 207 102. Linear equations 208 103. Equations reducible to the linear form 209 Section II. Equations of the First Order but not of the First Degree 104. Equations that can be resolved into component equations of the first degree . 210 105. Equations solvable for y 211 106. Equations solvable for x ........ 212 107. Clairaut's equation 213 108. Geometrical applications. Orthogonal trajectories . . .214 Section III. Equations of an Order Higher than the First 109. Equations of the form ^ = /(a;) 218 ■ dx" 110. Equations of the form ^ = /(?/) 218 111. Equations in which y appears in only two derivatives whose orders differ by unity 219 112. Equations of the second order with one variable absent . . 220 113. Linear equations. General properties. Complementary functions. Particular integral ... . . 222 114. The linear equation with constant coefficients and second member zero ,......••••. 223 XIV CONTENTS Art. Page 115. Case of the auxiliary equation having equal roots . . . 224 116. The homogeneous linear equation with the second member zero . 225 Examples on Chapter XIII . 227 APPENDIX Note A. A method of decomposing a rational fraction into its partial fractions .... .... 229 Note B. To find reduction formulae for ( x'"{a + bx'')P dx by integra^ tion by parts 231 Note C. To find reduction formulse for I sin" a; cos" x (ix by integra- tion by parts 233 Note D. A theorem in the infinitesimal calculus ..... 234 Note E. Further rules for the approximate determination of areas . 235 Note F. The fundamental theory of the planimeter .... 237 Note G. On integral curves . 240 1. Applications to mechanics ... . . 240 2. Applications in engineering and in electricity . . 242 3. The theory of the integraph ...... 244 Figures of some of the curves referred to in the examples . . 24() A short table of integrals . 240 Answere to the examples 263 Index 285 INTEGRAL CALCULUS CHAPTER I INTEGRATION A PROCESS OF SUMMATION 1. Uses of the integral calculus. Definition and sign of integration. The integral calculus can be used for two purposes, namely : (a) To find the sum of an infinitely large number of infinitesi- mals of the form f(x) dx ; (b) To find the function whose differential or whose differential coefiicient is given ; that is, to find an anti-differential or an anti- derivative. The integral calculus was invented in the course of an en- deavor to calculate the plane area bounded by curves. The area was supposed to be divided into an infinitely great number of infinitesimal parts, each part being called an element of the area ; and the sum of these parts was the area required. The process of finding this sum was called integration, a name which implies the combination of the small areas into a whole, and hence the sum itself was called the whole or the integral. From the point of view of the first of the purposes just indi- cated, integration may be defined as a process of summation. In many of the applications of the integral calculus, and, in particu- lar, in the larger number of those made by engineers, this is the definition to be taken. On the other hand, however, in many problems it is not a sum, but merely an anti-differential, that is required. For 'this purpose, integration may be defined as an operation which is the inverse of differentiation. It may at once be 1 2 INTEGBAL CALCULUS [Ch. I. stated that in the course of making a summation by means of the integral calculus it will be necessary to find the anti-differential of some function ; and it may also be said at this point, that the anti-differential can be shown to be the result of making a sum- mation. Each of the above definitions of integration can be de- rived from the other. These statements will be found verified in Arts. 4, 11, 13. In the differential calculus, the letter d is used as the symbol of differentiation, and ri/(.r) is read "the differential oif(x)." In the integral calculus the symbol of integration is * | , and I /(x) dx is read "the integral otf(x)dx." The signs d and | are signs of operations ; but they also indicate the results of the operations of differentiation and integration respectively on the functions that are written after them. The principal aims of this book are : (1) to explain how sum- mations of infinitesimals of the f orm /\.r) rf.v may be made; (2) to show how the anti-differentials of some particular functions may be obtained. 2. Illustrations of the summation of infinitesimals. Two simple illustrations of the summation of an infinite number of infinitely small quantities will now be given. They will help to familiarize the student with a certain geometrical principle and with the fundamental theorem of the integral calculus, which are set fortli in Arts. .3, 4. The method employed in these particular instances is identical with that used in the general case which follows them. * This is merely the long S, which was used as a sign of summatiou by the earlier writers, and meant " the sum of." The sign \ was first employed in 1075, and is due to Gottfried Wilhelm Leibniz (1646-1716), who inventeil the differential calculus independently of Newton. The word integral ap- peared first in a solution of James Bernoulli (10.54-1705), which was pub- lished in the Acta Eruditorum, Leipzig, in 1690. Leibniz had called the integral calculus calculus sniamatorius, but in 1696 the term calculus in- teyralis was agreed upon between Leibniz and John Bernoulli (1667-1748). 8ee Cajori, Histori/ uf Mathematics, pp. 221, 237. 1-2.] INTEGRATION A PHOCESS OF SlfMMATtON (a) Find the area betweea the line whose equation is y = mx, the X-axis, and the ordinates for which x = a, x = b. Let OL be the line y — mx\ let OA be equal to a, and OB to h, and draw the ordinates AP, BQ. It is required to find the area of APQ B. Divide the segment AB into n parts, each equal to L Y Qy Aas; and at the points of section A-^, A^, •••, erect ordinates A^P^, AiPi, ■■-, which meet OL in Pj, P^, ■■■. Through P, Pj, P^, •••, Q, draw lines parallel to the axis of x and intersecting the nearest ordinate on each side, as shown in Fig. 1, and produce PBi to meet BQ in C. It will first be shown that the area APQB is the limit of the sum of the areas of the rectangles PA^, P1A2, ■■•, when n, the number of equal divisions of AB, approaches infinity, or, what is the same thing, when Aa; approaches zero. The area APQB is greater than the sum of the " inner " rectangles PA^, PiA^, ■■■; and it is less than the sum of the "outer " rectangles AP^, AiP^, ■■-. The difference between the sum of the inner rectangles and the sum of the outer rectangles is equal to the sum of the small rec- tangles PPj, P1P2, •••• The latter sum is equal to JBiPiAx + SaPaAa;-! +B„Q^x•, that is, to (B,Pi + B2P2. + ■■■+B„Q)^x, which is CQ Aar. INTEGRAL CALU. 2 4 INTEGRAL CALCULUS [Ch. I- This may be briefly expressed, %APi - ^PA, = SPA = CQ Ax. When Ax is an infinitesimal, the second member of this equa- tion is also an infinitesimal of the first order ; therefore, when A.r is infinitely small the limit of the difference between the total areas of the inner and of the outer rectangles is zero. The area APQB lies between the total area of all of the inner and the total area of all of the outer rectangles. Hence, the area APQB is the limit both of the sum of the inner rectangles and of the sum of the outer rectangles as A.i- approaches zero. Each elementary rectangle has the area y A.v, that is mx Ax, since y = mx. The altitudes of the successive inner rectangles, going from A towards B, are ma, iii(a + Ax), in{a + 2Ax), •••, m(a+(n~l)Ax). Hence, A rea APQB = limit^^^„??i f a Aa; + (a + Aa;) Aa; + (a + 2 Aa;) Aa; + ■ • • +(a + n — 1 Ax) Axl * = limit^^^o"*^" +(« + Ax) + (a + 2 Aa!)+ ••• + (a + n — l Ax) \ Ax. Addition of the arithmetic series in brackets gives Area APQB = limit^^^o^^^^^ J2 a + (n - 1) Aa;} m (b-a) limit^j^o — i— — i \b + a — Ax\, since nAx=b — a, = m -- * The symbol Aa;=0 means " when Aa; approaches zero as a limit." It is due to the late Professor OUver of Cornell University. 2.] INTEGRATION A PROCESS OF SUMMATION In this example the element of area is obtained by taking a rectangle of altitude y, that is, mx, and width Ax, and then letting Ax become infinitesimal. The expression n = oo may be used instead of Ax = 0, since Ax -■ It may be noted in passing, that if the anti-differential of mxdx, namely — -, be taken, and b and o be substituted in turn for x, the difference between the resulting values will be the expression obtained above. (&) Find the area between the parabola y=oi?, the a^axis, and the ordinates for which x= a, x= b. Let QiOQ be the parabola y = a?; let OA be equal to a, and OB to h. Draw the ordinates AP, BQ. It is required to find the area APQB. Divide the segment AB into n parts each equal to Ax, and at the points of division A^, A^, •■-,, erect ordinates ^i^i, A^Pi, •■■■ Through P, Pi, P^, ■•■, draw lines parallel to the axis of X and intersecting the nearest ordinates on each side, as in Fig. 2. It can be shown, in the same way as in the previous illus- tration, that the area APQB is equal to the limit of the sum of 6 INTEGRAL CALCULUS [Ch. I. the rectangles PA^, P1A.2, ■■■, P„_i-B, when Ax approaches zero. This area will now be calculated. The area of any elementary rectangle is y Ax, that is a^ A.r, since y = a^- and the altitudes of the successive rectangles, going from A toward B, are a'', (a + Axy, (a + 2 Axf, ■■■. Hence, Area APQB = limit^^^„la'Ax + (a + Aa;)2Aa; +(a+2 Aa;/Aa; H + (a+ n — lAxfAxl = limit^^^of a2+(a. + Aa;)^ + (a + 2 Axf + + (a + ji— lA.r)2|Aa; = limit^^^oina'' + 2a Aa;(l+2 + 3H h w - 1) + (Axy(V+2' + 3'+:.+^^^\Ax. It is shown in algebra that the sum of the squares of the first n natural numbers, 1', 2\ 3^ •■•, n\ is ?K» + 1)(2m + 1) ^^^ 6 application of this result to the sum of squares in the second member of the last equation and the summation of the arith- metical series 1, 2, 3, ••■ in — 1), gives Area APQB = limit^^^„w Ax(a + Ax) — (a) — e„ Ax, f(a + Ax) Ax= (f>(a + 2 Ax) — (^ (a + Aa;) — ej Aa;, f(a + 2Ax)Ax= (a + 2Ax) — e^Ax, f(h — Ax) Ax = (b) — (b — Ax) — e„_i Aa;, in which each of the e's approaches zero when Aa; approaches zero. The sum of the first members of these equations is equal to the sum of the second members ; that is, \ /(a;)Aa; = (b)-(a)- a quantity which is not greater than (6 — a) E, and which ap- proaches zero when E ap- proaches zero, that is, when Aa; approaches zero. Therefore, on letting Aa; approach zero, there will be obtained. b /-6 Hence, the sum or integral, I f(x) dx, which is the sum of all the */ a values, infinite in number, that fix) dx takes as x varies by infinites- imal increments from a to b, is found by obtaining the anti-differ- ential fj> (x) of f{x) dx, and subtracting the value of (x) for x= a from its value for x=b. The following notation is used to indi- cate these operations : Jjix) dx= 4, (a;)1'= (x) -f-cT- Since the same c is used when a and 6 are substituted for x, this becomes £f(x)dx = axis, and the ordinates at a; = «, a; = b, the area may be regarded as representing the integral. It follows from the result (4) that a definite integral may be regarded as either : (1) The limit of the sum of an infinitely large number of infini- tesimal quantities of the form fix) dx taken between certain limits ; or, (2) The diiference of the values of the anti-differential of f(x)dx at each of these limits. If /(x) is any continuous function of x, f(x) dx has an anti-differ- ential.* However, the deduction of the anti-differential is often impossible, and in any case, is less simple and easy than the process of differentiation.f Many of the piractical applications of the integral calculus, such as finding areas, lengths of curves, volumes and surfaces of solids, centers of gravity, moments of inertia, mass, weight, etc., consist in making summations of infinitely small quantities. The integral calculus adds these infinitesimal quantities together and gives the result. It has been observed that in order to obtain the sum of infinitesimal areas, etc., the anti-differential of some function is required. Accordingly, a considerable part of any book on the integral calculus is devoted to the exposition of methods for obtaining the anti-differentials of functions which frequently appear in the process of solving practical problems. * The ti-uth of thLs statement, for all the ordinary functions, will appear in the sequel. A proof applicable to all forms of continuous functions is given in Picard, Traite W Analyse, t. I., No. 4. t The phrase "to And the anti-differential" means to deduce a, finite expression for the anti-differential in terms of the well-known mathematical functions. In cases in which the anti-differential cannot be thus obtained, an approximate value of the definite integral can be found by the methods dis- cussed in Arts. 84-88. A short inspection of these articles may be made now 4-5] INTEGRATION A PROCESS OF SUMMATION 13 5. Supplement to Art. 3. In proving the principle of Art. 3, the arc PQ in Fig. 3 was used. If the arc of the given curve has the form and position in Fig. 4, the proof of the principle is as set forth in Art. 3. If the arc has the form and position in \A A A \^_^ B T Pie. 6. Fig. 5, and thus has maximum and minimum values of the ordinate, the principle still holds. This can be seen by drawing the maximum and minimum ordinates that come between AP and BQ, and remarking that the principle holds for the several successive parts APP^Ai, AiP^P^A, •••. Suppose that the curve has the form and position in Fig. 6. The area of the part APA^ is the limit of the sum of the ele- mentary areas f(x) Ax when Ax approaches zero and x varies from OA to OAi; or, in other words, the area of APA^ is the limit of the sum of the elementary areas f{x) dx as x varies from OA to OAi- Similarly, the area of A-^TA^ is the limit of the sum of the elementary areas f(x) dx as x varies from OAi to OA2, and the area A2QB is the limit of the sum of the elementary areas f(x)dx as X varies from OA^ to OB. In APA^ and A^QB, the ordinates that represent the values oif(x) are positive, while in A1TA2 the ordinates are negative. Since x is taken as varying from left to right, dx is always positive. Accordingly, areas such as APAi, A2QB, which lie above the a^axis, have a positive sign, and areas such as A1TA2, which lie below the x-axis, have a neg- ative sign. This example shows that in the case of a curve that crosses the ar-axis, the method of summation by means of the integral calculus gives the algebraic sum of the areas that lie 14 INTEGRAL CALCULUS [Ch- 1. between the curve and the x-axis, the areas above the a^axis being given a positive sign, and those below receiving a nega- tive sign. If the total absolute area between the curve and the axis of X is required, the portions APA,, A,TA„ A,QB, should be found separately. Note. If n is a constant not equal to - 1, the anti-differential of u"du is ""'*" • for, differentiation of the latter gives u'^du. »+ 1 Ex. 1. Find the area between the curve whose equation is y = x', the a;-axis and the ordiuates for which x = 1, x = 4. By Art. 4, the area required = | x^dx ii-'l 2_5_6_ , = 63J units of area. Ex. 2. Find the area between the parabola 2j/ = 5x^, the x-axis and the ordinates for which x = 2, x = 5. .4ns. 97^ square units. Ex. 3. Find the area between the line j/ = 4 x, the x-axis and the ordi- nates for which x = 2, x = 11. Ans. 234 square units. Ex. 4. Find the area between the parabola 2y = 3x^, the x-axls and the ordinates for which x = — 3, x = 5. Ans. 76 square units. Ex. 5. Find the area between the line y = 5x, the x-axis and the ordinate for which X = 2. Ans. 10 square units. Ex. 6. Find the area between the line y = 5x, the x-axis and the ordinates for whicli X = — 2, X = 2. Ans. 0. 6. Geometrical representation of an integral. It is necessary to perceive clearly that a definite integral, whether it be the sum of an infinite number of infinitesimal elements of area, length, volume, surface, mass, force, work, etc., can be represented graphi- cally by an area. Eor instance, in order to represent the definite integral J .f(x)dx, 5-7.] INTEGRATION A PROCESS OF SUMMATION 15 choose a pair of rectangular axes, plot the curve whose equation is y=f(^), and draw the ordinates for x = a, x = b. It has been shown in Art. 4 that the area between this curve, the a>axis, and these ordinates has the value of the definite integral above. Hence, this area can represent the integral. This does not mean that the area is equal to the integral, for the integral may be a length, a volume, etc. The area can be taken to represent the integral, because the number that indicates the area is equal to the number that indicates the value of the integral. That an integral may be represented geometrically by an area is at the foundation of some important theorems and applications of the integral calculus. 7. Properties of definite integrals. In Art. 4 it was shown that if the definite integral, | f{x) dx = (b) — (b) — <^(a). Hence, the algebraic sign of a definite inte- gral is changed by an interchange of the limits of integration. Q>) f''f(x)dx^jy{x)dx+jj{x)dx. Let the curve whose equation is y = f(x) be drawn ; and 16 INTEGRAL CALCULUS [Ch. I. let ordinates AP, BQ, CR be erected at the points for which x= a, X = b, x= c. Since area APQB = area APBC + area CBQB, C''f(:x)dx= C''f(x)dx+ Cfix) %Ja Ua *Jc dx. It does not matter whether c is X between a and h or not. For, sup- pose OC = c', and draw the ordi- nate OR' ; then area APQB = area APR'C - area BQR'O ; and hence, ( f(x)d.c= I f(x)dx— | f(x)dx, ■ Jn Ja J^ fix) dx-{- I f-(x) dx. Therefore a given definite integral may be broken up into any number of similar definite integrals that differ only in the limits between which integration is to be performed. (c) Construct APQB as in (6) to represent the definite integral r f{x) dx. Then r f{x) dx = area APQB = area of a rectangle whose height CR is greater than AP and less than BQ, and whose base is AB, =^AB-CR = (b-a)f(c), OG being equal to c. 7.] INTEGRATION A PROCESS OF SUMMATION 17 Jf(x) dx 6 — a The function /(c) is called the mean value of f(x) for values of X that vary continuously from a to b. This mean value may be defined to be equal to the height of a rectangle which has a base equal to 6 — a and an area that is equivalent to the value of the integral. The subject of mean values is discussed further in Arts. 76, 77. CHAPTER II INTEGRATION THE INVERSE OF DIFFERENTIATION 8. Integration the inverse of differentiation. In Art. 1, two definitions of integration were indicated, namely : (a) Integration is a process of summation ; (6) Integration is an operation which is the inverse of differen- tiation. The first definition was discussed in Chapter I. In this and tlie next following article, integration will be considered from the point of view of the second definition. The differential calculus is in part concerned A^dth finding the differential or the derivative of a given function. On the other hand, the integral calculus is in part concerned with finding the function when its differential or its derivative is given. If a function be given, the differential calculus affords a means of deducing the rate of increase of the function per unit increase of the independent variable. If this rate of increase of a function be known, the integral calculus affords a means of finding the function. Ex.1. A curve whose equation isy = ix^ is given ; and the rate of increase of the ordinate per unit increase of the abscissa is required. Since y =4x'^, ^ = 8x. Ox This means that the ordinate at a point whose abscissa is x is increasing 8 x times as fast a,s the abscissa. If tliis rate of increase remains uniform, tlie ordinate will receive an increase of 8 x when the abscissa is increased by unity. Tliis determines the direction of the curve at the point. 18 8.] INTEGRATION INVERSE OF DIFFERENTIATION 19 On the other hand, suppose that at any point on a curve, it is known that ax and let the equation of the curve be required. It evidently follows from this equation that in which c is an arbitrary constant. The constant c can receive any one of an infinite number of values ; and hence, the number of curves that satisfy the given condition is infinite. If an additional condition be imposed, for example, that the curve pass through the point (2, 3), then c will have a definite corresponding value. For, since the point (2, 3) is on the required curve, 3 = 4.22 + c, and accordingly, c = — 13. Hence, the equation of the curve that satisfies both of the conditions above given is y = ix^-13. Ex. 2. In the case of a body falling from rest under the action of gravity, the distance s through which it falls in { seconds is a constant, approximately 16 times f^ feet ; find the velocity at any instant. Here, s = l6fi, and hence, — = 32 * ; at that is, the velocity * in feet per second at the end of t seconds is 32 1. On the other hand, suppose it is known that in the case of a body falling from rest, the velocity in feet per second is 32 multiplied by the time in seconds since motion began. Let the corresponding relation between the distance and the time be required. ds Here, it is known that — =;: 32 f. dt * If a body moves in a straight line through a distance As in a time AJ, and Lf its average velocity be denoted by v, As At As At approaches zero. As also approaches zero, and the velocity approaches ds the definite limiting value — • INTEGRAL CALB. — 3 20 INTEGRAL CALCULUS [Ch. II. It is obvious' that tlie solution of this simple differential equation is (1) s = 16 «2 + c, in which c is an arbitrary constant. Tliis result is indefinite. By the condi- tions of the question, however, s = when t — 0. Hence, substituting in (1), = + c ; whence, c = 0, and ' s = 16 J2 is the definite solution. The distance through which the body falls can also be deter- mined by the method of summation employed in the first chapter. Let the number 32 be denoted by g. The distance passed over in any time is equal to the product of the average velocity during that time and the time. The time t may be divided into n equal intervals Af, so that t = n At The velocity at the beginning of the rth interval is (/■ — l)g Af, and at the end of the interval is ;•(/ \t. Hence, the distance passed over in the interval lies between (r - l)g(My and rg{Aty. On finding similar limits for the distance passed over in the case of each of the intervals and adding, it will be found that the total distance passed over lies between l0 + l+2+... + (n- l)]9'(Ai)^ and [1 -f 2 -f- • ■ • -f- n]g(My; that is, summing these arithmetical series, the distance passed over lies between !Li!^^(AO^and?i(^^(AO=. Since Ai = -, the distance lies between n 9l'-9^and2f + ^. 2 2n 2 2«' and the distance is the common limit of these two expressions, when Ai approaches zero, that is, when n approaches iniinity. Hence, s = ^gf. 8.] INTEGRATION INVERHE OF DIFFERENTIATION 21 Sometimes the anti-differential (or the anti-derivative) of a func- tion is wanted for its own sake alone, as in the illustrations given above ; and sometimes it is desired for further ends, as, for example, in the process of making a summation by means of the integral calculus in Art. 4. The anti-differential is called the integral, the process of finding it is called integration, and the symbol of integration is the sign | . Thus, if the differential of ,^ix) is /(a;) dx, which is expressed by d(x):=f(x)dx, (1) then Cf(x) dx = (x). (2) Equation (1) may be read " the differential of {x)." For brevity, the latter may be read "the integral oi.f(x)dx is <^(a;)." * Memory of the fundamental formulae of differentiation will carry one far in the integral calculus. For instance, since dx* is ia^dx, |4a^da; is x*; since cZ sin a; is cosxdx, (cosxdx is sin X. The beginner will see the necessity of having ready com- mand of the formulae for differentiation, since they will be employed in the inverse process of integration.! Differentiation * The origin of the terms integral, integration, and of the sign i has been given in Art. 1. Instead of the sign | , the symbols d'^ and i?-i are sometimes employed: thus, d~^f(x)dx, which is read "the anti-differential of /(x) dx, ' ' and D'\f(x) , which is read, ' ' the anti-derivative of /(x) . " In the case of tlie second definition of integration, the use of the symbols d~'^, 2)-i, is more logical than the use of | . The latter sign is, however, fiimly estab- lished in this connection. It may be remarked that the differential is more frequently written than is the derivative of a function. t The expressions Ix^dx, d~''-(x^dx), Z)~i(x2) are equivalent. The in- verse process of integration is not always practically possible (see Art. 4). Art. SI may be referred to for examples of differentials whose integrals can- not be expressed in a finite form. 22 INTEGRAL CALCULUS [Ch. II. of both members of (2) gives d I /(ar) dx = dcji (x), whence, by (1), =f(x)dx. Therefore, d neutralizes the effect of | . It will be shown in the next article that | dct>(x) may have values different from (x). 9. Indefinite integral. Constant of integration. Since d(xl^ + c) is 4 x' dx, c being any constant, I ix^dx is x* + c. The integral given in Art. 8 comes from this on making c zero. But c may be given any other value that does not involve a;. Hence, lAafdx is indefinite so far as an arbitrary added constant is concerned. In general, if d(x) + c, (2) in which c is any constant; for differentiation of the members of (2) shows that f(x)dx= d4>(x). Hence, the integral of a given differential is indefinite so far as an arbitrary added constant is concerned^ Illustrations have been seen in the preceding articles. It should be noted that the indefiniteness does not extend to terms that contain x. In other words, a given differential can have an infinite number of integrals that correspond to the infi- nite number of values that an arbitrary constant can take, but any two of these integrals differ only by a constant. For instance, /' (x + l)dx = — + x + Ci. But on substituting z for a; + 1, and consequently, dz for da;, j (a; + l)da;= J zdz = -+C2= '- ^ ^ + 0; = - + a; + - + C;. 8-10.] iNTt:GttATtOif INVMEtSE of mPFEtttlNTtATtON 23 These two integrals agree in the terms that contain x. When an integration is performed, the arbitrary constant should be indicated in the result ; or, if not indicated, it should be under- stood to be there. The second member of (2) is usually called the indefinite integral of f{x) dx, and c is said to be the constant of integration. When the constant of integration has an arbitrary value, that is, when no definite value has been assigned to it, the integral is called also a general integral ; on the other hand, when the constant of integration is given a particular value, the integral is said to be a particular integral. For instance, the general integral (and indefinite integral) of si'dx is ^a* + c in which c is arbitrary. A particular integral of off'dx is obtained by giving c any one of an infinity of possible values, say 6, — 5, ^. Thus ^ a;'' + 6, \a^ — 5, ^^^ + ^ are particular integrals. In practice the value of the constant may be determined by the special conditions of the problem. 10. Geometrical meaning of the arbitrary constant of integration. If S=^'^*)' (^) then y=\ F'(x) dx, that is, y = F(x) + c, ' (2) in which c is an arbitrary constant of integration. Suppose that c is given particular values, say 8, — 3, etc. ; and let the curves whose equations are y = Fix) + S,^ y = F(x)-3,l etc., etc., :ves have i is, the same direction, for the same value of x. Also, for any be drawn. All of these curves have the same value of -^; that dx 24 INTEGRAL CALCULUS [Ch. II. two of the curves the difference in the lengths of their ordinates remains the same for all values of a;. Tor example, for any value of x, the dif- ference betvireen the lengths of the ordinates of the two curves whose equations are given in (3) is 8 - (- 3), or 11. Hence, all the curves, whose equations are of the form (2), thus differ- ing merely in the c's, can be obtained by moving any one of the curves vertically up or down. The particular value assigned to c merely determines the position of the curve with respect to the X-axis, and has nothing to do with its form. Fig. 9 illustrates this. 11. Relation between the indefinite and the definite integral. In Art. 1 it has been seen that if d(x) =f{x)dx, the sum or inte- gral of f{.v) dx for all values of x from x = a to a; = 6, satisfies the relation £f(x)dx={a). (1) If the upper limit be variable and be denoted by x, jj (x) dx = ^{x)-ct> (a) . (2) If the lower limit a be arbitrary, — (a) may be represented by an arbitrary constant c, and (2) becomes But J f(x) dx = (x) -f- c. J f(x) dx= (x) + c. (3) (4) 10-12.] INTEGRATION i:^VERSE OF DIFFERENTIATION 25 Hence, an indefinite integral is an integral whose upper limit is the variable and whose lower limit is arbitrary. The first member of (4) may be considered an abbreviation for the first member of (3). The indefinite integral may therefore be re- garded as obtained by a process of summation. 12. Examples that involve anti-differentials. This article is in- serted for the purpose of giving simple, typical examples of a practical kind, in which anti-differentials are required for pur- poses other than that of summation. These illustrations will also involve the determination of constants. In many applica^ tions of the calculus, two hinds of constants must be distinguished, namely, those which are constants of integration, and those which are given distances, angles, forces, etc. ; for example, the constant g, in Ex. 2, Art. 8, and h, h, k, a, in Ex. 2 below. Eec- tangular coordinates are used in the following exercises. Ex. 1. Determine the equation of the curve at every point of which the tangent lias the slope J. Determine the equation of the curve whicla passes through the point (2, 3) and also satisiies the former condition. Tlie slope of a curve y =/(x) at any point (x, y) is -^.* Hence, by the given condition, (1) ^ = 1. ^^ dx 2 Adopting the differential form, dy = ^dx, and integrating, (2) y = lx + c, tlie equation of a straight line. Now c, the arbitrary constant of integration, which in this case represents the intercept of the line on tlie //-axis, can take an infinite number of values. The first condition is therefore satisfied by each and all of the parallel lines of slope ^. If, in addition, the line is required to pass through the point (2, 3), then x = 2, J/ = 3, satisfy (2), and 3 = | • 2 -|- c. From this, c = 2. Hence, the curve that satisfies both of the given conditions is the line whose equation is y = ix + 2. * By the slope of a curve at any point is meant the tangent of the angle that the tangent line to the curve at the point malies with the a^axis. 26 INTEGRAL CALCULUS [Cii. II. Ex. 2. Determine the equation of the curve that shall have a constant subnormal. Also, determine the curve which has a constant subnormal and passes through the two given points (o, ft). (6. k)\ and find the length of its constant subnormal. Let A, B, be the given points (o, ft), (6, k), and let Pbe any point (x, y) on the curve. Suppose that PT is the tangent at P, and PiV the normal. Put the constant subnormal ilfiV equal to a. Since the angle a = angle 9 (see Fig. 10), their sides being respectively perpendicular, tan a = tan ; that is, (1) dy _a dx~ y Putting this in the differential form, (2) ydy = n dx, M N Fig. 10. 2 + C' = ax + c", (3) 2 ~ ax + c. and integrating both sides. whence, in which c', c", are the constants of integration, and c denotes c" — c'. Equation (3) is the equation of a parabola, and it includes an infinite number of parabolas, one for each of the infinite number of values that the arbitrary constant c can have. The particular curve which passes through the points (o, ft) (6, k), and has a constant subnormal is also required. Since the coordinates of these points must satisfy (3), it follows that ft2 and F ab + c. 12-13.] INTEGRATION INVERSE OF niFFKIiENTIATIOy 27 These equations suffice to determine c and the length a of the constant subnormal. On solving them, it is found that ^ 26 Hence, the equation of the second curve required is 6!/2 = (A;2 - /l2) X + 6ft2. In the differential calculus it is shown that the length of the subnormal is y —■ The first condition might have been expressed immediately by the equation which is equivalent to (1) and (2). Ex. 3. Find the curve whose subtangent is constant and equal to a. De- termine the curve so that it shall pass through the point (0, 1). Ex. 4. Find the curve for which the length of the subnormal is propor- tional to (say k times) the length of the abscissa. 13. Another derivation of the integration formula for an area. In Arts. 3, 4, it was shown that the area inchided between the curve y =f(x), the a;-axis, and the ordinates for x = a, x = b is the limit of the sum of the infinitely large number of infinites- imal quantities /(.i:) dx, which are successively obtained as x varies continuously from a to b, and this limit was represented by the definite integral I f(x) dx. The area can also be derived by means of the second defi- nition of integration. Let CPQ be an arc of the curve whose equation is y = f(x), and let OA = a, OB = b. Draw the ordinates AP, BQ. Take any point S on the a>axis at a distance x from 0, and draw the ordinate SL whose length is f(x). Let z denote the area of OCLS. If x or OS is increased by ST, which is equal to \x, and the ordinate TM be drawn, the area 2 will be increased by the area SLMT. This increment will Fig. 11. 28 INTEGRAL CALCULUS [Ch. II. be denoted by Az. Draw LR parallel to the a>axis, and complete the rectangle LRMV in which LR = Aa; and RM= ^y. The increment of the area, Az = SLMT = rectangle SLRT + area LRM = SLRT+ an area less than the rectangle LRMV = f(x) \x + area less than Ay ■ Ax. Hence, Az — =/(«) + something less than Ay. Ax When A.v approaches zero, Ay also approaches zero ; and hence, in the limit, |=/(«^); (1) that is, -— = y. (2) ax Equation (2) means that the numerical value of the differential coefficient with respect to the abscissa, of the area between a curve, the axes of coordinates, and an ordinate, is the same as the numerical value of this ordinate of the curve. Equations (1) and .(2) written in the differential form give the differential of this area, namely, dz =/(«) dx, and dz = ydx. (3) Finding the anti-differentials in (3) gives as the area OCLS, I dx = ^(x) + c, (4) in which <^(a;) is the anti-derivative of f{x) and c is an arbitrary ■■=ffix), 13-14.] INTEGRATION INVERSE OF DIFFERENTIATION 29 constant. If the area is measured from the y-axis, the area is zero when x is zero. Hence, substituting these values in (4) = <^(0) + c, whence, c = — <^(0), and (4) becomes z = (f>(x) — <^(0). Hence, area of OOPA = (^(a) - ^(0). Similarly, area of OOQB = (b) - <#.(0). Since APQB = OCQB - OGPA, it follows that area APQB = <^(6) — <^(a). The expression in the second member is the same that was found for the area in Art. 4 by means of the first definition of integration. If the area is measured, not from the ?/-axis, but from another fixed vertical line, say the ordinate at a; = m, the derivation of equations (1) and (2) is the same as given above. In this case, the area is zero when a; = m, and hence, = <^(m) + c. From this, c = — <^(m). The value of c in (4) thus depends solely upon the fixed ordi- nate from which the area is measured. 14. A new meaning of y in the curve whose equation is y = f{x). Derived curves. It has been seen in the differential calculus that in the case of a curve whose equation is y =f(x), at any point on the curve the slope of the curve is — , the differential dx coefficient of its ordinate with respect to its abscissa. Art. 13, with equations (2) and (4), shows that at any point of a curve the length of the ordinate y is the differential coefficient with respect to the abscissa, of the area bounded by the curve, the 30 INTEGRAL CALCULUS tCH. H- axis of X, a fixed ordinate, and the ordinate at the point * There- fore, " if we wish to make a graphic picture of any function and its derivative, we can represent the function either by the ordi- nate 2/ of a curve or by its area, while its derivative will then be represented by its slope or ordinate respectively. If we are most interested in the function, we usually employ the former method (in which the ordinate represents the function) ; if in its derim- tive, the latter (in which the ordinate represents the derivative). That is, we usually like to use the ordinate to represent the main variable under consideration." t For instance, suppose it is necessary to represent the function fix). Let the curve be drawn whose equation is 2/ =/(*)• (1) At any point {x, y) on the curve, the ordinate y represents the value of the function for the corresponding value of x ; and the slope — represents the rate of change of the function compared with the rate of change of the variable x. Now let the curve be drawn whose equation is y=f'(F), (2) in which /'(a;) denotes ^' • At any point {x, y) on this curve, * The remaining part of tliis article is not necessary for tlie articles that follow. However, it may be useful for the beginner to read it, because it may help to strengthen his grasp on the fundamental principles of the integral calculus. t Irving Fisher, A brief introduction to the Infinitesimal Calculus designed ('Siicrinllij to aid in rcadlnrj mathematical economics and statistics, Art. 80. Some readers may be intprested in an application of the principle quoteil above. Vrofp.ssdr Fisher continues: "Jevons, in. his Theory uf Political L'conomy, used the abscissa x to represent commodity, and the area z to represent its total utility, so that its ordinate y represented ' marginal utility' (i.e. the differential quotient of total utility with reference to com- modity). Anspitz and Lichen, on the other hand, in their UntersucJmngen uber die Thenrii- des PnHses, represent total utility by the ordinate and marginal utility by the slope of their curve." H.] INTEGRATION INVERSE OF DIFFERENTIATION 31 the numerical measure of the ordinate is the same as that of the slope of the first curve at the point having tlie same abscissa. Hence, an ordinate at any point of the second curve represents the rate of change of the function /(a;) compared with the rate of change of the variable x at this point. Also, the area between the second curve, the axes, and the ordinate at (x, y) is that is rf(x)dx, Hence, the area of the curve y=f'(x) bounded as described above plus a constant quantity /(O) can represent the function For example, suppose that the function is paf + 4. That is, f(x)=px' + 4:, and f'(x) = 2px. The parabola y = pa? + 4, and the line y = 2px are shown in Fig. 12. At any point M on the avaxis, the ordinates MP, MQ Y ^ --^y / 1 / ^^ \s^ 4'' \ \ '-' ^"-^ / _.,-''' o Fig. 12. are drawn to these curves. The ordinate MP represents the function for x= OM; and the slope at P represents the rate of change of the function when x = OM. The ordinate MQ is equal (numerically) to the slope at P; and hence, it also repre- 32 INTEGRAL CALCULUS [Ch. II. sents the rate of change of the function when x = OM. More- over, since area OQjW= | 2pxdx = px', Jo the function f{x) for x=OM is represented by the areaOQilf+4. Had the function been ps? (shown by the dotted curve), the area OQ,M would exactly represent the function. To recapitulate : In the case of a function f(x), if the curve y=m, (1) and its first derived curve y = /'(«), (2) be drawn, the rate of change of the function for any value of x is represented equally well by the slope of the first curve and by Y Fig. 13. the ordinate of the derived curve for that value of x; and the function itself for any value of a; is represented equally well by the corresponding ordinate of the primary curve and by the area of the derived curve increased by the constant quantity /(O). 14-15.] INTEGRATION INVERSE OF DIFFERENTIATION 33 The derived curve (2) is called the "curve of slopes" of the first curve. Two of these curves are shown in Fig. 13. The horizontal scale is the same for both curves ; but the ordinates on the original curve represent lengths while the ordinates on the derived represent tangents of angles. At a point at which the original curve has a maximum or a minimum ordinate, the slope is zero; and hence, the corresponding ordinate on the derived curve is zero. Conversely, when the derived curve crosses the a>axis, the corresponding ordinate of the original curve is a maximum or a minimum. 15. Integral curves. Let the curve whose equation is 2/ = /(«') (1) be drawn. Suppose that the anti-derivative of f(x) is ^{x) ; and draw the curve whose equation is y=Cf(x)dx, (2) that is, y = {0), or, briefly, y = F(x). (3) The curve whose equation is (2) or (3) is called the first inte- gral curve of the curve (1). It is evident that ^a = ^ = /(x). (4) dx dx ■'^ '' ^ ■' The following important properties can be deduced from equa- tions (1), (2), (4). (a) For the same abscissa x, the number that indicates the length of the ordinate of the first integral curve is the same as the number that indicates the area between the original curve, the axes, and ordinate for this abscissa. Therefore, the ordinates of the first integral curve can represent the areas of the original «urve bounded as above described. 34 INTEGRAL CALCULUS [Ch. II. (b) For the same abscissa x, the number that indicates the slope of the first integral curve is the same as the number that indicates the length of the ordinate of the original curve. There- fore the ordinates of the original curve can represent the slopes of the first integral curve. KlG. u. Example. The line whose equation is y=px has for its first integrat curve the parabola whose equation is y= \ 2}xdx, ^0 that is, f=p- a;" At any point M on the a;-axis, OM being equal to x^, say, erect the ordinates MP, JfPi. to the line and the parabola. The same number, namely ^^^, indicates both the length of the ordinate J/Pi and the area 0PM; and the same number, namely ^).r„ indi- cates both the length of the ordinate MP and the slope of the tangent at Pj. This is true for the pair of ordinates erected at every point on the a--axis. In like manner the curve whose equation is (2) has a first integral curve. The latter is called the second integral curve for the curve of equation (1). This second integral curve has a 15-16.] INTEGRATION INVERSE OF DIFFERENTIATION 35 first integral curve which is called the third integral curve of (1), and so on. There is thus a series of successive integral curves for any given curve. For instance, the second integral curve of the line y=px is the curve whose equation is '=So^ that is, y=P-^ o 7? 2' This curve is shown in the figure. The subject of successive integral curves has very important applications in problems in mechanics and engineering. Accordingly, an exposition of their properties and uses is given in Chapter XII. 16. Summary. This and the preceding chapter have been con- cerned with showing by statement and examples that integration may be regarded in two ways : (1) As a process of summation, in which | f(x) dx denotes the limit of the sum indicated by y /(a;)/\x, when Aa; approaches zero; '■='^ (2) As an operation which is the inverse of differentiation, in which j fix) dx denotes d~'^[f(x) dx'] or D~'^f{x) ; that is, denotes the anti-differential of /(a;) dx, or, what is the same thing, the anti- derivative -of f(x). It may be remarked that the rules of integration are all derived from the latter point of view. Both of these conceptions of integration are employed in problems in geometry, mechanics, and other subjects. The first view of integration is necessary to a clear understanding of the application of the integral calculus to the solution of certain problems; and, on the other hand, the second view is necessary to a clear understanding of the use of the calculus in the solution of certain other problems. INTEGRAL CALC. 4 CHAPTER III FUNDAMENTAL RULES AND METHODS OF INTEGRATION 17. In Chapters I. and II., the two purposes of integration were set forth; and definitions of integration based upon these purposes were given with illustrative examples. Relations be- tween the definitions were also pointed out, particularly in Arts. 11, 13. It was also shown that in the process of making an integration, whatever the object may be, it is necessary to find an anti-differential or an anti-derivative of some function. A general method of differentiation is given in the differential calculus. Unfortunately, no general method for the inverse pro- cess of integration exists. It is necessary to derive a rule for the integration of each function. The formulae of integration are derived or disclosed by falling back upon our knowledge of the rules of differentiation. In fact, the first simple rules, given in Art. IS, are merely directions for retracing the steps taken in differentiation. The inverse operation of finding an integral is, in general, much more difficult than the direct operation of find- ing a differential or a derivative. This chapter gives an exposition of the fundamental rules and methods employed in integration. One or more of these rules and methods will come into play in every case in which integra- tion is required. 18. Fundamental integrals. Following is a list of fundamental formulae of integration derived from the fundamental formulae of differentiation. They can be verified by differentiation, as 36 17-18.] RULES AND METHODS OF INTEGRATION 37 indicated in the first of the set. An additional list is given in Art. 22. Every integrable form* is reducible to one or more of the integrals given in these two lists. The student should have ready command of these formulae for two reasons : first, so that he may be able to integrate these forms immediately ; and second, so that he may know the forms at which to aim in reducing com- plicated functions. The functions to be integrated will not usually present themselves in terms of these simple, immediately integrable expressions ; and therefore a considerable part of this book is taken up with algebraic and trigonometric transforma- tions showing how to reduce given functions to these forms. In the following formulae, u denotes any function of a single independent variable. J«' ^au = ^ + c. n + l in which n has any constant value, excepting — 1. The case in which m = — 1 is given in II. Differentiation of each member of I. with respect to u gives ?t" du. n.t I -— =logt«-l-Co = logM-|-logC = logCM. The different ways in which the arbitrary constant of inte- gration can appear in this form, may be noted. ; T" * "An integrable form " here means a function whose integral can he ex- pressed in a finite form which Involves only algehraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions. J,/— 1+1 1 M-i du = |-c = - + c=x + c. Nevertheless, / — 1 -f- 1 _ . I M-i du can be derived directly by means of I. For, on putting — — + ci -' n + l for c, which is allowable by Art. 9, \u" du = — — ^ -|- Ci. Now — — ^ Q J n+l n+l = - when n = — l. Evaluation of this indeterminate form by the method of ■^ the differential calculus gives, differentiating numerator and denominator as to n, M"+'logM, that is, logM. Hence \u-^du=\ogu +cx. 38 INTEGRAL CALCULUS [Ch. III. III. ia>'du = -^^ + c. J log a IV. Te" rtM = e" + c. V. ( sill u («) +•••}<«*' for differentiation of each member of A gives f{x)+Fix) + ^ Ex 7 C ^"'"'^^ r «^(?« r x^dx ! ' ' J a + 6x»' J 4 + 3 1)8' J 5 - + 6x» J 4 + 3 1)8 J 5 - 2 x* Ex.8. c^!^«_, f-^-, r^^ J4-v/TTr52 J 9 + 4x2 Jl 4-n/I - «2 J 9 + 4 x2 J 16 + 25 2/2 ■^ 20. Integration aided by a change of the independent variable. Integration can often be facilitated by a convenient change of the independent variable. For instance, if f{x) dx is not imme- diately integrable, it may be possible to change the independent variable from x to t, the relation between x and t being, say X = iji{t), so that f(x) dx is thereby put into a form F{f) dt which can be easily integrated. Experience and practice afford the only means of determining the substitutions that will be helpful in particular cases. The actual substitution of the new variable may often be conveniently omitted, as in Exs. 1, 2, 3, below. Ex. 1. \(x + a)"dx. On putting x + a = t, dx = dt ; and the given Integral becomes J n + l re + 1 Since dx = d(x + a), the given integral may also be written j (x + a)"d(x + a), and x + a being regarded as the variable m, the integral is (^ + a)""^^ + c, as before. n+l J:2 INTEGRAL CALCULUS [Ch. HI. Ex. 2. f seo2 (5 - 2 k) c?x. If 5 — 2 X = «, da; = — J (?J ; and J seo2 (5 - 2 k) (?a; = - J ("seeded* = - J tan « + c = -itan(5-2a;)+ c. Since £fx = — ^(?(5 — 2a;), the integral may be written also fseo^ (5 - 2 x) dx = - i f sec2 (5-2 x)(i (5 - 2 x) = — J tan (5 — 2 x) + c. Ex. 3. le'+'^dx. If a + 6x = ^ dx=-dt, and 6 f e^+'^cZx = - (e'dt =-e' + c = -€'+'•' + c. Otherwise : since dx = -d(a + bx) , b r^o+6i^ = - (e'+'"d (a + bx) = -e«+»» + c. Ex.4, r 12x^-4x + 5 ^ J 4 x3 - 2 x2 + 5 X - 10 On putting 4 x" - 2 x^ + 5 x - 10 = «, it follows that (12 x'^ ~ i x + 5)dx = dt, and f . .^^"".'T^"!"^^ ,„ fe=C^ = logi + c=Jog(4x3-2x''+5x-10) + c. ^4x» — 2x^+5x — 10 Jt Note. 7^ Jfte expression under the sign of integration is a fraction whose numerator is the differential of the denominator, the integral is the logarithm of the denominator. Ex. 5. foos^'^-K sin^x If sinx = t, cosxdx = dt ; and Jcosxcix Cdt 1 , , 1 sin^x ~ J t^" bt^ " '-■ ■= ■ )£* ' ' 5sin^x Otherwise: C<^^^^dx ^Cdj^mx) ^ L+c. J sin''x J (sinx)^ Ssin^x The necessity of learning to recognize /orms readily will be apparent. 20.] RULES AND METHODS OF INTEGRATION 43 Ex. 6. rl^i^^. ■ J(x+1)< If X + 1 = z, dx = dz, X = z — 1, then r ^Mx ^r(3-l)3^,^r/l_3 3_1X^^ J(x + 1)* J 0i J\z ^ z^ z*) = logz + ?--^ + -L + c = z 2 z2 3 33 logz+^ iz^-9z+2 Qz" =-<->-^*1^ t"+.. Ex. 7. •^ (e« + 1)* If e'+l = z, e'dx = dz, e' = z — 1 ; and h p2x \e'+l)i ^ zi =J(,.- z~i) dz = f z« - ^ z* + c = ^ z* (3 z - 7) Ex. 8. (ix + a)^dx, ({x + a:iidx, f-^, f— ^, f^^^. r(2 + 3x)^(Za;, ("(3 -7x)^cZx. Ex. 9. rcos(x + a)dx, I sec2(x+a)cZx, | — „ . ^ „ — , I sin(a+6x)(i(; J J J cos2(4 — 3 x) J Ex.10, fcos-dx, Ce2+5x(ix, (e'^dx, (^^dx. J 2 J J Jsin^x ;^Ex. 11. rx(a + x)*dx, f ^'^^ (a + 6x)^ VExi2 (■ ^^^^ , rcos(\ogx)dx^ r de '' ■ J(l+x2)tan-ix J X \in^li\ Ex.13. C(a + 6«)^dz, C\/(a + 6x)2(ix, ("^7==== J J J V(a + 6«/)* 44 INTEGRAL CALCULUS [Ch. III. Ex. 14. C , ^^ (Put 2 a; - 1 = 4 0.) J \/l5 + 4x-4x2 <2_ Ex. 16. f °°sx(?x ^p^^ sinx = 2.) j"(sin«e - 3 sin^S + 4 sin^fl + llsine + 2)cosed9, ("(tan'f - 7 tan^ ,() + 2 tan ,f + 9) sec^ ,f d,^. 21. Integration by parts. Two universal formulse of integra- tion were given in Art. 19. A third formula of this kind will now be discussed. Differentiation shows that, u and v being any functions of x, d , ^ du , dv — (uv) = V h u — dx dx dx This may be written also in the differential form, d(uv) = v(—]dx + u(—]dx, \dxj \dxj or more simply, d (uv) = vdu + udv, (1) in which, du = — dx, and dv= — dx. dx dx Equation (1) becomes on transposition, udv= d (uv) — V du. Integration of both members of this equation gives C. \udv = uv — \v du. Equation C may be used as a formula for integrating u dv when the integral of v du can be found. This method of integration, commonly called " integration by parts," may be adopted when f(x)dx is not immediately integrable, but can be resolved into two factors, say u and dv, such that the integrals of dv and v du are easily obtained. The procedure is as follows : \f(x)dx = I Mdv; whence by C, =uv— (v du. .20-21.] nULES ANt) METSODS OF tNTBQtlATtON 46 No general rule can be given for choosing the factors m and di). Facility in using formula C can be obtained only by practice. This formula has a greater importance and a wider application, than any other in the integral calculus. The following examples will show how it may be employed. Ex. 1. Integrate x sin x dx. Let u = x, dv = sin x dx ; then, du = dx, v = — cos x. Application of C gives \ X sin xdx = — X cos x + \ cos x dx ; = — x cos cc + sin a: + c. Ex. 2. Integrate \ogxdx. Let u = log X, dv = dx; then, du-~, v = x. X Therefore by C, i\ogxdx = a log a; — \x — = X log X — X + c. Ex. 3. Find Xxe^dx. Let u = e', dv = xdx ; then, du = e'dx, « = J x^. The formula gives I xe' dx = JxV — i) x'^e'dx. But x'^e'dx is not so simple for integration as xe'dx. This indicates that a different choice of factors should have been made. On putting u = x, dv = e'dx, du = dx, v = e", and formula C now gives I xe' dx = xe' — {e'dx = xe" — eF + c. 46 INTEGRAL CALCULUS [Ch. III. Ex. 4. Find ix'e'dx. Let u = x', dv = e'dx ; then du = Z x^dx, v = e*. Hence, I a;V dx = a;V — 3 \x^' dx. To flud I a;^e* da;, put u = 7?, dv = e'dx; then du = 2xdx, v = e" ; and I x^e" dx = a;'2e^ — ^ixe" dx. By Ex. 3, \xe' dx = xe" — e" + c. Hence, combining tlie results, (x^e'dx = e'(x^ -Sx'^ + 6x - 6) + c. This is an example in which several successive operations of the same kind are required in order to effect the integration. Many such examples will be met, and usually a formula called " a formula of reduction " will be found for integrating them. "Integration by parts" is of great use in deducing these formulae of reduction. In order to avoid making mistakes in cases like Ex. 4, a good plan is to write down the successive steps in the integration clearly, without putting in the intermediate work, which can be kept in another place. Thus : I x^e" dx = x^e'^ "^ \ ^^^' ^^ — x^e' — 3 [x%^ — •2{xe' dx'] = ofie'' — 3 [K^e"! _ 2 (xe"^ - e')] + c = e'{x^-Sx'' + 6x-6) + c. Ex. 5. \sin~^xdx. Ex.10. Xx^sinxdx. Ex. 6. icot-'^xdx. Ex.11, ix'cosxdx. Ex. 7. \za'dz. Ex. 12. ix\,a.a-^xdx. Ex. 8. (x-a'dx. Ex.13. ({Xogxydx. . 9. (tan-ixda;. Ex.14. T cosfflogsinflde. Ex 21-22.] RULES AND METHODS OF INTEGRATION 47 Ex.15. I sec'^ailogtauxda;. Ex.17. \xe""dx. Ex.16. (x^(logxydx. Ex.18. (x"'\ogxdx. 22. Additional standard forms. Some fundamental integrals which often appear are collected together in the following list. Their derivation will be found in the next article. XV. J tan udu = log sec u + C. XVI. ( cot udu = log sin u + C. XVII. j'secMdM = logtanf^ + |Wc. XVIII. f cosec udu = log tan ^ + C XIX. C *^^ = sin-i- + C = - cos-i - + C. XX. f_^=ltan-i^ + C = -lcot-i^+C'. XXI, r ^ = - sec-i ^ + C = - - cosec-i ^ + C Xxir. r ^"^ = vers-i -+C = - covers^ - + C XXIII. C— ^??^— = J-log*^^^+ C = itanh-i-+C'. J u'^ -a^ 2a u + a a a XXIV. f "'^ =Iog(^ + VM^ + VvF+a' 2 « , , ... du + dz and, by composition, — Therefore, i , = i u + z 'du + dz ^u^ + a^ -^ «* + « = log (?t + «) + c = log (m + Vu^ + a') + c ^ U + Vw" + a/' , • 1,-lM . = log ^ — 1- Ci = sinh ' - + Cj. 50 INTEGRAL CALCULUS [Ch. III. The latter result can be derived also in the following way : On putting m = az, C /^ = C-^= = sinh-'^ + c. -f V^F+a' -^ V2' + 1 = sinh~' - + c. a Formula XXV. Similarly, on putting u^ — a^ = z% { —^ — = logfa+VM^-a^+c = log — ■ h C2 = cosh ' - + C2. Or, putting u = az, r /" = r ^^ = cosh-'z + c = cosh-' « + c. •^ Vm' - a? -^ V«' - 1 « Ex. 1. Find ("Va^ -x^cZx. Integrating by parts, let Then (J« = ^ dx, v = x. and ' -'-' -'•' '-' "■' ' ' ^"^'^^ r Va^ - x2 fte = X Va2 - x^ + f — Va2 _ a;2 Since Va^ _ a;2 = " ^ ^° . it follows that — — = "' - Va^- x^. Va2-x-^ Va2-x2 Va-^-x2 Hence, f Vn'^ - x^ (Zx = xVa'^ - x'^ + a^ f ^'■'' - f V«2 _ a;2 dx. From this, on transposing the last integral to the first member, 23.] RULES AND METHODS OF INTEGRATION 51 Ex. 2. { ^^ = C ^ •^ V24 + 10 a; - x"^ -^ V4!) - (oi' - 10 x + 25) -' V7''-fx-5^2 V 7 y V7'' - (X - 5) r ^ ^r •^ Vx2 + 8x + 52 -^ Ex. 3 ' "^ - ' <^* V(x + 4)2 + 36 = log (X + 4 + Vx2 + 8x + 52). Ex. 4. f-^-^^ =(■ ^ = -^tan-i^ Jx2 + 6x + 12 J(x + 3)2 + 3 V.3 + 3 (X + 3)^ + 3 V3 V3 Ex. 5. C ^ =r ^ -lie^ (x + 3)-2 Jx^+6x+5 J(x + 3)^-4 4 °(x + 3)+2 Ex. 6 4 ^x + 5 (2x Ex. 8. f ^^'^^ Ex. 17. f- (^X VS X* + 2 x2 - 1 -^ t3,n ax Ex. 9. r ^ Ex. 18. foot (ax + 6)(?x. J_3 + 4x-x2 J r 5& Ex. 19. f_^-^. Ex. 10. V4 X — x^ F^ 11 r '^^ Ex.20, f g^_r=r '^(^-2) 1 3^f^" Jx-4x + 8L J(x-2)244j T7v 19 r '^^ Ex. 21. f ^ Ex. 13. ffi tan 3 x (fc. Ex. 22. j* (sec ix + iydx. Ex. 14. r — idx _ j,^ 23. r_ (^X V3-5x2 -^ (x-o)V(x-o)2-62 INTEGEAL CALC. 5 52 INTEGRAL CALCULUS [Ch. III. Ex. 24. r ''^ Ex. 29. f ^— Ex. 25. f^^. Ex.30. (l±^^d 13 Ex. 26. jxyy/x^-y^dx. j,^ g^ T d 2bx + c Ex. 27- ^^^ + 2V3 Ex.32, f-v/^+lcfe. ^ .,, J 'a; - 1 . 28. I - Ex. "- ' '^ Vis a; — 6a;2 [Kationalize the numerator.] 24. Integration of a total differential. It has been shown in the differential calculus, that if u=fix,y), (1) X, y, being independent variables, the total differential of u is equal to the sum of its partial differentials with respect to x and y. That is, du = —dx + ~dy. (2) dx dy " ^ ^ It will be remembered that when differentiation is performed with respect to x, y is regarded as constant, and when differenti- ation is performed with respect to y, x is regarded as constant. Suppose that a differential with respect to two independent variables is given, namely, Pdx+Qdy, (3) in which P and Q are functions of x and y. The anti-differential of (3) is required. Not every function (3) that may be written at random has an anti-differential. Hence, it is necessary to de- termine whether an anti-differential of (3) exists or not, before trying to find it. It has been shown in the differential calculus that if u and its first and second partial derivatives with regard to X, y are continuous functions of x, y, then. dy dx dx dy (4) 23-24.] RULES AND METHODS OF INTEGRATION 53 If (3) has an integral, say u, then, du=Pdx+Qdy, (5) in which P = — , (6) and Q = |^. (7) Differentiation of both members of (6) and (7) with respect to y and x, respectively, gives 3P a^M by dy dx dx dx dy Hence, by (4), |=f. (8) Therefore, if (3) has an integral, relation (8) holds between the coefficients P, Q, and the differential (3) is then said to be an exact differential. Conversely, it can be shown that if relation (8) holds, the differential (3) has an integral. For the present the latter proposition may be assumed to be true.* The condition (8) is called the criterion of the integrability of the differen- tial (3). Suppose that the coefficients P, Q satisfy the test (8), then there is a function u which satisfies equation (6). Since Pdx can have been derived only from the terms that contain x, integration of the second member of (5) with respect to x gives fpdx +c, in which c denotes any expression not involving x. * For proof, see Introductory Course in Differential Equations, Art. 12, by D. A. Murray (Longmans, Green & Co.). 54 INTEGRAL CALCULUS [Ch. III. Now Qdy has been derived from all the terms of u that contain y. Some of these terms may contain x also ; and if so, they have been discovered already in | Pdx. Therefore, the remaining terms of u that do not contain x will be found by integrating with respect to y the terms of Q dy that do not contain x. Hence, the following rule: Integrate Pdx as if y were constant; inte- grate, as if x were constant, the terms in Qdy that do not contain x; add the results and the arbitrary constant of integration. Ex. 1. Integrate ydx + xdy. Here F=v, Q = x\ hence ^ = 1, ^=1, and thus, criterion (8) is " ^ dy dx satisfied. Also, i Pdx = \ydx = xy; and there are not any terms in Q dy without x. Hence the integral is xy + c. Ex. 2. Integrate ydx — xdy. sp^,,dq^_i,: Here P = y, Q = — x: hence -^r— = 1, -^ = — 1, and the criterion is not ^ dy dx satisfied. Tlierefore an integral of the given expression does not exist. Ex. 3. Integrate (x^ — 4 xy — 2 y'') dx + (y^ — ixy — 2 x^) dy. Ex. 4. Integrate (a^ — 2xy — j/^) dx — (x + y)^ dy. Ex. 5. Integrate (2 ax + by + g) dx + (2ay + bx + e) dy. 25. Summary. The directions so far given for obtaining the indefinite integral, of f(x) dx may be summarized as follows : (1) Memorize the fundamental formulae of integration given in Arts. 18 and 213. (2) Acquire familiarity with the application of the principle of substitution, or change of the independent variable, discussed in Art. 20. (3) Use the first and second universal formulae of integration, A, B, given in Art. 19. (4) Learn to apply with ease the third universal formula of integration, namely, the formula for integration by parts given in Art. 21. 24-25.] RULES AND METHODS OF INTEGRATION 55 EXAMPLES ON CHAPTER III 1. f ac2 x°-' dx, r (m + '0 a;"+"-^ dx, i{m + n) u"'+''+' dv, f^x" 'dx, (a;W'z^''-^dz. Z. r('x2 + l-_3x3+4V^' fV - 2/b'<2)/. Jx + 2 J z-2 J3W-I 4. Find the functions whose differential coefficients are ? -i 2 , x", x ", x~", v'^ - 3 ajjT + 4av ^. 5. Find the anti-differentials of (sec2 B + oosec-2 9) de, (3 cos 2 - 5 sin 3 0) d0, ^"^ '^ '^'^ . ^ a + 6 cos ^ 6. Find the anti-derivatives of (—^ + —^\ COSX-I--J— , ilogx, —^ log (ax -1-6). \y — a y+a/ cos^x x ax + b 7. Evaluate the following definite integrals : C(x~^ + 5z'')dx, pcos4e(Ze, (^cosixdx, C^ Ji Jo Jo Ji 2 P ^'^^ , 4:Cha,n2 0d0. Jo Vl - 4x2 -'« I 1 8. Ipg25f 5^dx, C'^'J^ C (e<-+e')dx, C(e-+e--)dx T 9. Ce-^'''+^xdx, r '^^ . PcotelogsinSdff, Jo VI - x2 -^^ l+x^ 56 INTEGRAL CALCULUS [Ch. III. 10. C'l ^ + ^ \dx, P(sm20 + cos30)m(m - 1) •■•3.2 log x+ (-l)"-™!]. 21. Show that J g^x" dx = x^eF - m i e'^x"-^ dx = e'[x" - mx™-! + ni(m - l)x'»-2 + (-l)'»-im(m-l) ...3.2.x + (-!)-». ml]. 25.] RULES AND METHODS OF INTEGRATION bl 22. \ (sec X + cosec x)"^ dx. 23. C ^ . J 1 + cot e 24. j" '^^ Va2 + 62 - x2 26. Evaluate 2 T log tan ■ cosec 2 fl d9 + log cot B \ cosec 2 9 dfl. Jit ./ff 26. j (a tan 9 sec 9 + 6) cot $ de. 27. C ^ C ^ J V2 + 2 a; - a;^' J V- 16 - lOsc-K^ 28. flog (sc + Vx2 - o2) ^. 29. flog (x + Va;2 + a^) '^^ ■ ^ Va;2 + a2 30. ie^'°°'''*dft -. C a sec" 9 + 6 cosec" g ,„ gg /" sec a: tan a: (?g J tan 9 + cot 9 ' J tan" a; — 2 (Ja; J a; on i '^^ vers-'"' V^da;. ' J e« — ( .^ C dx 33. ^y/^fl±^^dx. *"■ J VS^^^:^' /. j„ ., ,. sina;(Za; "*• f^^-T- (Put =» = «'•) "• f . ^ , a;2 — 5 xi J sin- vcos x 38. f^^ 42. C ^_ J sin a; + cos x J . f ^^ . . 43. f. ^ cos" — sin" J 36 aa;^ + 6x + c (Ja; 37. cos" - sin" J Vaa;" + 6a; + c dx f(3£+il^^±Iaxis, and two ordinates for which a; = a, x=h, is ex- pressed by Xf{x)dx. It has also been shown that this area can be evaluated by- finding the indefinite integral of f{x)dx, substituting h and a in turn for x in the indefinite integral, and taking the differ- ence between the results of the two substitutions. L Y P.^ 3i Xr— B S M O BJ^U-—^ A 9 D n " '"' Q —X Fig. 15. Ex. 1. Pind the area bounded by the parabola whose equation is y^ = i:ax, the axis of x, and the ordinate at a; = x-y. Also find the area between the parabola y^ = Qx, the axis of x, and the ordinates for which x = i, x = 9. Let QOG be the parabola whose equation is y^ = 4:ax. Take 0M= Xi, 0A = A, 0D = 9; erect the ordinates MP, AB, DC. Suppose that two ordinates US, VT are drawn at a distance dx apart. The element of area, which is the area of any infinitesimal rectangle like B8TV, is ydx. The area required in the first case is equal to the sum of the areas of all such rectangles, infinite in number, that are between OY and PM; that is, be- tween the limits zero and Xi for x. Hence, area of 0PM = \ ^ydx. Jo First of all, y must be expressed in terms of x. means of the equation of the curve, from which y = ±2 a^x^. This can be done by 1675), professor of mathematics at the College of Prance in Paris, Blaise Pascal (1623-1662), John Wallis (1616-1703), Savilian professor of geometry at Oxford, considered an area to be made up of infinitely small rectangles, and applied the principle to the determination of the areas of parabolic curves. The Prench geometers found the formula for the area between the curve y = x'", the axis of x, and any ordinate x = h when to is a positive integer. Wallis found the area when m is negative or fractional. This was before the development of the calculus by Leibniz and Newton. 60 INTEGRAL CALCULUS [Ch. IV. (The positive sign denotes an ordinate above the x-axis, the negative, one below.) Hence, area of 0PM = | 2 a^x^dx Jo that is. area of 0PM = two thirds of the area of the circumscrib- ing rectangle OLPM. The area OPQ = 2 0PM = two thirds the area of the rectangle LPQR. In the second case : area .45 CD =V ydx = 3 Cx^dx = 3 [f 35^ + c]9 = 38. If the unit of length is an inch, the area of ABCD is 38 square inches. Ex. 2. Find the area between the* curve y^ = i ax, the axis of y, and the line whose equation is ^ = 6. Y A B 1 ^^ 1 / ^^ i^,y) X V Fig. 16. In this case it is more convenient to take for the element of area the infinitesimal rectangle indicated in the figure. The element of area is thus * dy ; and area OAB = \ xdy Jo 'i = .fc* 4a 12 a* 27.] OEOMBTBIOAL APPLICATIONS 61 Ex. 3. Find the area of the ellipse — + ^ = 1, The area required is four times the area of the quadrant A OB. An ele- ment of area is the area of an infini- tesimal rectangle IISTV, namely dx. The sum of all these elements from O to ^ is expressed hy I ydx. From the given equation, Pia. IT. y = ±- Va^ - x', a in which the positive sign denotes an ordinate above the ,x-axis, and the negative sign, an ordinate be-' low. Hence, area of ellipse = 4 OAB = i\ ydx Jo =:4:^CVa^-xUx; aJo which by Ex. 1, Art. 23, = i* [2 Va^ - x'^ + ^sin-i^ + cT a L2 2 a Jo = irab. ' If a = 6, the ellipse is a circle whose area is wa^. Find the area included between the ordinates for which x = 1, % = 4, the curve, and the axis of x. Area PQMN= Vydx = ^C Va' - x^ dx = 6r2V^^3^4.«!sin-i«-HcT al2 2 a Ji = -|2Va2-16 - iVS2^n: + ^(sin-i^ - sin-i^^ |. If the semi-axes are 5 and 3, area PQMN =1{Q-VQ + ^ (sin-i ^ - sin-i J)} = f {6 - 2.454) -I- ^ (.927 - .201) = 3.778. 62 INTEGRAL CALCULUS [Ch. IV. Fis. 18. Ex. 4. Find the area between the curve whose equation is the axis of x, and the ordinates for which x = — 2,x=7. 7 The area required = \y dx- -2 7 = ^^ f (^^ - 9a;2 + 23a; - I5)dx -2 = iilix^ -Sx^ + y-x^ - 15x + c'f Further remark on this example may be instructive. On putting ^ = 0, the intersections of the curve and the x-axis are seen to be at the points for which x = l, 3, 5. That is, referring to the figure, 0C= 1, OD = 3, 0E = 5. Area APC =(ydx=- ^. This area appears with a negative sign, since the ordinates are negative in APC because it is below the x-axis. Area CHD=( ydx = + i. the sign coming out positive since C3D is above the it-axis. and AxesLDLE= \ ydx=—^; ajea EQB = \ ydx=+S. The area required = area APC + area CUD + area DLE + area EQB = — ff , as obtained before The absolute area =J^--|-J + ^ + 3 = 12||. 27-28.] GEOMETRICAL APPLICATIONS 63 This is an example of the principle indicated in Art. 5, namely, that when the area between a curve, the x-axis, and any two ordinates, is found by inte- gration, this area is really the sum of component areas, those above the X-axis being affected with the positive sign, and those below the x-axis with the negative sign. The next example will also serve to illustrate this. Ex. 5. Find the area between a semi-undulation of the curve y = sinx and the x-axis. The curve crosses the x-axis at x = 0, x = ir, x = 2 tt, etc. Areaof^BC=l ydx=\ sinxdx = [— cosx + c] =2. area ABODE = ( "ydx^ \' "sin x dx = 0. Jo Jo The total area, regardless of sign, is 4. Y But Fig. 19. 28. Precautions to be taken in finding areas by integration. The method of finding areas which has been described in the last article can be used immediately and with full confidence in the case of a curve y=f(x), only when the limits a and b are finite, and the function f(x) is continuous and one-valued for values of X between a and b, and does not become infinite for any value of X between a and b. Special care must be taken in cases in which any one of the conditions just mentioned does not hold. While, in some of these cases, the application of the method of Art. 27 will give true results, in other cases it will give results that are altogether erroneous. A few examples are given below in order to emphasize the necessity of caution. Ex. 1. There is a double value for y in the parabola y^ = 4: ax. This was considered in Ex. 1, Art. 27. Ex. 2. Eind the area included between the parabola (y — x — 6)^ = x, the axes of coordinates, and the line x = 6. 64 INTEGRAL CALCULUS [Ch. IV. In this case y = a; ± Va + 5 ; and thus to each value of x belongs two values of y. The ambiguity can be removed by defining more exactly what area is meant. If the area OBPM is desired, the value of y corresponding to each value of a between and 5 is a; — VS + 5. Hence, / Q 3 / r\ / \ ,'" ''' / ,>' \ B Ic:;;^ P ^^ 5 area OBPM = f {x-y/x + b)dx Jo =[f-i"'-^"-]: V M Pio. 20. If the area OB QM is desired, the value of y corresponding to each value of x be- tween and 5 is a; + Vx + 5 ; and hence, area OBQM = \ (a; + v^ + 5) dx Jo = V- + ^^■ If the area PBQ, between the curve and the line a; = 5, had been required, it would have been necessary first to determine the areas OBPM, OBQM. Area PBQ = area OBQM- area OBPM; = i^-VE. Another way of finding the area of PBQ is the following. Let TS be any infinitesimal strip of width dx parallel to the !/-axis. Evidently, TS is the difference of the values of y that correspond to x = V. Hence, denoting these values of y by yi, j/a, area PBQ = \ (yi — yi) dx Jo = (" {(a; + Vx + 5)-(a;-\^ + 5)}da; = r 2y/xdx Jo Ex. 3. Eind the area included between the witch w = — and its X2 + c(2 asymptote. The asymptote is the axis of x, and hence, the limits of integra- tion are + oo and — co. In this case it is allowable to use Infinite limits. For, on finding the area OPQM between the curve, the axes, and an ordi- nate at distance x from the origin, 28.] QUOidEtRICAL APPLICATIONS area OPQM =\ ydx Jo _ C' a^dx Jo a;2 + a2 = ranan-i? + cT L a Jo a 65 M If the ordinate MQ be made to move away from the origin towards the right, that is, if the upper limit x increases continuously, then tan-i - increases continuously, and approaches - as a limit. Hence, I. x^ + a''~ 2 represents the true value of the area to the right of the y-a.xis. Since the curve is symmetrical with respect to the !/-axis, the area required is double this, namely, va^. Ex. 4. Find the area included between the curve y^ (x^ — a^Y = 8x\ the K-axis, and the asymptote x = a. In this case y = ^ To every value of x corresponds a real value (a;2 - a2)|- of y ; but, when x = a, 2^ is infinite. Therefore a special examination is re- quired. For values of x less than a, however, y is finite. Then, for x a and x = 3 a, area M'P'qx= f— =^ 2 :r dx :6a3 ■ a2)3 3 (x2 - a'^y As X diminishes and approaches a, this area approaches 6 a^ ; and hence, the area between the infinite ordinate at x = a, and the ordinate at x = 3a, is 6 (13. Hence, the total area between the curve, the x-axis, and the ordi- nate at X = 3a, is 3 a^ + 6 o^, that is 9a'- The same resvilt is obtained when ( — Jo 2 xdx is evaluated in the ordi- (x2 - a2)S 28-29.] GEOMETRICAL AMPLICATIONS 67 nary way ; and thus, the integi-ation fonnula for the area holds good in this case, although f{x) becomes infinite for a value of x between the limits of integration. Ex. 6. Find the area included between the curve y(x — ay = 1, the axes, and the ordinate a; = 2 a. Immediate application of the integration formula gives for the area. C^ " dx _ r _ 1 ^-|2° -_ 2 Jo {x-ay L x-a Jo ~ a Fig. 23. But, /(x), which is the length of the ordinate y, becomes infinite for x = a; and, if an investigation be made similar to that carried out in Exs. 4, 5, ifrwill be found that the area is infinite. For, OM being equal to x, area OMPB = C — ^— ■ = -^ -• Jo (x — ay a — x a It is evident, that as x increases from to a, the area increases from to 00. Consequently, the area between the curve, the axes, and the ordinate at a; = a is infinite. Similarly, it may be shown that the area between the curve, the a^axis, and the ordinates at a; = a, a; = 2 a, is infinite. Therefore the total area required is infinite. Hence, the integration formula for tlie ely, f "^ •^ Jo {x-a area, namely, I — ^^ — i fails in this particular case in which /(x) be- Jo (a; — ay comes infinite for a value of x between the limits of integration. This con- clusion may be compared with that in Ex. 5. 29. Precautions to be taken in evaluating definite integrals. It has been shown in Art. 6, that any definite integral, say j f{x)dx, may be graphically represented by the area between the curve y =/(«), the axis of x, and the ordinates a,tx = a,x = b. Hence, INTEGRAL CALC. — 6 68 INTEGUAL CALCULUS [Ch. IV. the statement at the beginning of Art. 28, and the precautions described in that article, must be applied when any definite inte- gral C f{x)dx is under consideration. EXAMPLES IN AREAS.* 1. Find by integration the areas of tlie triangles bounded by the co- ordinate axes, and each of the following lines : (a) 7x + 52/-35=0; (6) lSx-y-l2 =0. 2. Find the areas of the triangles bounded by the z-axis, and (o) the lines 7a;-3i/-21 = 0, x = -5; (6) the lines 5x4-6^ + 15 = 0, x = -l. 8. Find the areas of the triangles bounded by the y-axis, and (a) the lines 9x + 4j/-6 = 0, j/ = l; (6) the lines 2x + y + 8 = 0, y=-i- 4. Find the area of the figure bounded by the axis of abscissas, the curve y = x2 + X -1- 1, and the ordinates corresponding to the abscissas 2, and 3. 5. So for the ourye j/ = x* + 4 x^ + 2 x^ + 3 between the abscissas 1, 2. 6. Find the area of the figure cut offl from the curve ?/ = (x + 1) (x -f- 2) by the x-axis. 7. Find the area included between the semi-cubical parabola y'^ = ofi and the line x = 4. 8. Find the area included between the semi-cubical parabola 'f = x^, the {/-axis, and the line !/ = 4. 9. Find the area included by the parabola ?/2 = — 4 x, and the line x = — 1. 10. Find the area included by the parabola x^ -|- 12 j/ = 0, and the line J/ = -3. 11. Find the total area included by the curve y = x', and the line y = 2x. 12. Find the area of the first quadrant of the circle x^ + j/^ = j^. 13. Find the area intercepted between the coordinate axes and the parab- ola X* + 2/^ = a^. 14. Find the area included between the hyperbola xy = k^, the x-axis, and the ordinates at x = a, x = 6. * Figures of some of the curves referred to in examples throughout the book are given in the Appendix. 29-30.] GEOMETRICAL APPLICATIONS 69 ^30. Volumes of solids of revolution. Let PQ be an arc of a curve ■whose equation is y =f(x). Draw the ordinates AP, BQ, and let OA=^a, OB=b. The vohime of the solid PQML generated by the revolution of APQB about the a;-axis is re- quired. On the revolution of PQ each point in the arc PQ will describe a circle. Suppose that AB is divided into ii equal parts Aa;, and let OQi= x, Q1Q2 = Aa;. Construct the rectangles PiQit P2Q1 ^s indicated in the figure, and suppose that they have revolved about OX with APQB. It is evident that the volume of each plate, such as PiPg-^z-^D of the solid of revolution is less than the volume of the corre- sponding exterior cylinder gen- erated by the revolution of the rectangle Q1R1P2Q2 about the avaxis, and greater than the vol- ume of the corresponding interior cylinder generated by the revolution of the rectangle Q1P1R2Q2 about the a>axis. Now, the volume of the cylinder generated by ^1^1^262 Fio. 24. = iry^^x = ^[f(x)yAx; and the volume of the cylinder generated by QiBiPiQi = TrP2Q2^X = 7r[f(x + Ax)YAx. Hence, tt lf(x)fAx < PiAiVs^^i < tt [/(a; + Ax)y Aa;. Suppose that PQML is divided into n plates, such as P1P2N2N1, one plate corresponding to each segment Arc of AB ; and suppose 70 INTEGRAL CALCULUS [Ch. IV. also that the interior and exterior cylinders corresponding to each of these plates are constructed. Then, on taking the sum of all the interior- cylinders, and the sum of all the exterior cylinders, and the sum of all the plates PiP^N^Ni, the latter sum being the volume required, x = b 3; = & V T [/(a;)]2A.-v < PQML kS^'^ [f(x+Ax)YAx. x=a x=a As Aa; approaches zero, the sum of the exterior cylinders ap- proaches equality to the sum of the interior cylinders. The difference between these sums is at the most an infinitesimal of the first order when Ax is an infinitesimal, and accordingly has zero as its limit. Therefore, since the volume required always lies between these sums, volume PQiUfiy = limit / w \^f(x)y Ax ; x = a that is, volume PQML = {\ [/(a?)]2 da;. The element of volume is -rr \_f(x)fdx; this is usually written TTif^dx, since y —f(x). This value of the element may readily be deduced from the figure on supposing that Q1Q2 is an infinitesimal distance. If an arc of y =/(«) between the points for which y = c and y = d revolves about the y-axis, it can be shown in a similar way that the element of volume is -n-x' dy, and that the total volume generated by the revolution is rrj^ <^dy. Before integrating it will be neces- sary to express x^ in terms of y. 30.] GEOMETRICAL APPLICATIONS 71 Ex. 1. Find the volume of the right cone generated by revolving about the a; axis the line joining the origin and the point (A, a). Let M be the point (h, a). The equation of OM\s ax = hy. The element of volume is wy"^ dx. Hence, volume OMN = ^1 2/' = TT I — Jo } dx ,2j.2 'O ft2 3 dx. Fie. 26. This may be interpreted : the volume of the right circular cone OMN is equal to one third the area of the base by its altitude. Ex. 2. Find the volume of the cone generated by the same line on revolv- ing about the !/-axis. In this case, the element of volume is Tx"^ dy. Hence, volume OMN = TT j x'^dy Jo =^r^dy Jo a' ■ah?- ''dx 3 Ex. -3. Find the volume of the solid generated by revolving the arc of the Fig. 27. parabola j/^ = ipx between the origin and the point for vrhich x = Xi, about the a^axis. In this case, volume OPP\ = t i y^ Jo = t\ ipxdx Jo = 2 irpxi'^ ; or, since yi^ = ipxi, OPP, = ^- it Hence, the volume is one half the volume of the circumscribing cylinder. Fig. 2S. 72 INTEGRAL CALCULUS [Ch. rv. Ex. 4. Find the volume generated by revolving the arc in Ex. 3 about the y-axis. In this case, i X Fig. 29. volume OPQ = t I x'dy Jo or, since j/i^ = 4^zi, volume OPQ = -j ti/iXi^. Hence, the volume required is one fifth the volume of the cylinder of base PQ and height CO. Ex. 5. Eind the volume of the solid generated by the revolution about the a;-axis of the arc of the curve ^ = (x + 1) (a; + 2) between the points whose abscissas are 1, 2. Ex. 6. Find the volume of the cone generated by the revolution about the' a;-axis of the parts of each of the following lines intercepted between the axes : (a) 2 X + 2/ = 10 ; (c) 4 a; - 5 1/ + 3 = ; (6) 7x + 2y + S = 0; (d) 3z-Sy = 6. Ex. 7. Find the volume of the cone generated by the revolution about the S/-axis of the parts of each of the following lines intercepted between the axes : la) ix + 3y = 6; (c) 5 a; - 7 y + 35 = ; lb) 3x-4y = 6; (d)2x + 6j/ + 9 = 0. Ex. 8. Find the volume of revolution about the x-axis of the arcs of the following curves between the assigned limits : (a) ^2 = x3, X = 0, X = 2 ; (6) (a^ + x^) y* = a*, x = 0, x = a. Ex. 9. Find the volume of the solid generated by the revolution about the X-axis of the curve y^ = ex from the origin to the point whose abscissa is Xi. Ex. 10. Find the volume of the solid generated by the revolution of the same arc as in Ex. 9 about the y-axis. Ex. 11. Find the volume of the solid generated by the revolution about the y-&xis of an arc of the curve in Ex. 9 from the origin to y = yi. Ex. 12. Find the volume of the prolate spheroid generated by the revolu- 0-2 y2 tion of the ellipse — |- 2- = 1 about the x-axis. 30-31.] GEOMETRICAL APPLICATIONS 73 31. On the graphical representation of a definite integral. In Arts. 4, 6, attention has been drawn to the principle that any definite integral, whether it denotes volume, length, surface, force, mass, work, etc., may be graphically represented by an area.* A simple illustration may put this in a clearer light. Ex. Find the volume of the right cone generated by revolving about the a-axis the line drawn from the origin to the point (4, 1). Let P be the point (4, 1), and let POQ be the cone of revolution. The equation of OP is iy = x. Hence, vol. PO§=j'V2/2(?a; The volume is thus | t cubic units of the same kind as the hnear unit employed. In order to represent this volume graphically, draw the curve OSB whose equation is 16 a;^ 16 x^ being the function of x under the sign of integration in (1 ) ; and draw the ordinate CB at a; = 4. The area iJOC graphically represents the volume POQ. For, area BOO = ( ydx Jo "* WX^ , -r 16 -dot, (2) Equations (1) and (2) show that the number of cubic units which indi- cates the volume of POQ is the same as the number of square units which indicates the area of JtOG. In the same way, if the ordinate NH be drawn at any point N, for which x = a, say, it can be shown that -^-g ira^ denotes both the number of cubic units in the cone MOL and the number of square * On account of this property the process of integration was called by Newton and the earlier writers "the method of quadratures." 74 INTEGRAL CALCULUS [Ch. IV. units in the area HON. It is thiis apparent that the number of cubic units in LMPQ is the same as the number of square units in NHBC, namely ^7r(64:-a*). Hence, vol. POQ : vol. MOL : vol. LMPQ = area BOC : area HON : area NHBC. (3) If the curve y :=-^ tmrx^ be drawn, the numbers which indicate the areas will be m times the numbers which indicate the volumes of the correspond- ing sections of the cone. But the ratio of any two right sections of the cone will be the same as the ratio of the two corresponding areas, and proportion (3) will still hold. The curve y = -^ mirx'^ can therefore be used to represent the volume. It is sometimes well to use a multiplier m for the sake of con- venience in plotting the curve that will graphically represent the integral. Note. If the first integral curve (see Art. 15) of OHB, namely, y= r^x2 = i-«3 Jo 16 48 be drawn, its ordinates represent both the areas of the segments of OHB and the volumes of the segments of the cone POQ measured from O. 32. Derivation of the equations of certain curves. Oftentimes, when a curve is described by some property belonging to it, the formal analytic statement of the property involves differential coefficients. In these cases the derivation of the equation of the curve consists in finding a relation between the coordinates which will be free from differentials. Examples of this have been given in Art. 12. A few additional simple instances are introduced here. In the larger number of cases the derivation of the equa- tion of the curve will require a greater knowledge of differential equations than the student possesses at this stage ; and hence further problems of this kind will be deferred until Chap. Xlll. Ex. 1. Determine the curve whose subtangent is n times the abscissa of the point of contact. Find the particular curve which passes through the point (5, 4). Let (x, y) be any point on the curve. The subtangent is y—. By the given condition, ^ y^=nx. dy This may be written, — = — -• X y 31-32.] GEOMETRICAL APPLICATIONS lb Integrating, log c + log a: = n log j/ ; whence, y^ = ex. (X) All the curves obtained by varying c, satisfy the given condition. If one of the curves passes through the point (5, 4), for instance, 4" = 5 c. (2) Substitution in (1) of the value of c from (2) gives 5 ^" = 4"x, as the equation of the particular curve through (5, 4). What curves have the given property for re = l? n = 2? re = |? n = \1 n = f? Ex. 2. Find the curves in which the polar subnormal is proportional to (is k times) the sine of the vectorial angle. What particular curves pass through the point (0, 2 tt) ? dr The polar subnormal is — By the given condition, d6 — = /fc sin e. de Integrating, r = c — k cos 8. For the curve that passes through (0, 2 7r), = c — ft; whence c = k. Hence, the equation of the particular curve required is r = i;(l — cosS), the equation of the cardioid. Ex. 3. Determine the curve in which the subtangent is n times the sub- normal ; and find the particular curve that passes through the point (2, 3). Ex. 4. Determine the curve in which the length of the subnormal is pro- portional to the square of the ordinate. Ex. 5. Determine the curve in which the subnormal is proportional to (is k times) the nth power of the abscissa. Ex. 6. Find the curve in which, for any point, the length of the polar subtangent is proportional to (is k times) the length of the radius vector. Ex. 7. Find the curve in which the angle between the radius vector and the tangent at any point is n times the vectorial angle. What is the curve when^ji = 1 ? when « = J ? 76 INTEGRAL CALCULUS [Ch. IV. EXAMPLES ON CHAPTER IV 1. rind the area of the figure bounded by the curve a;* + ax^ + a'^x,^ + bh/ = 0, the X-axis, and the ordinates at a; = 0, a; = a. 5 3 8. Find the area inclosed by the curve xy^ = j/^ -f 2 j/^ and the lines x = 0,y = 0,y = l. 3. Find the area included between the parabolas y^ = iax and x'' = 4 ay. X X 4. Find the area included between the catenary y = ^(^e' + e "), the axes of coordinates, and the line x = c. 6. In the logarithmic curve y = C"" prove that the area between the curve, the axis of x, and any two ordinates is proportional to the difference between the ordinates. 6. Find the area included between the curve y = — - — •, and the line 1+x^ 7. Find the area bounded by the curve y = x' + ax^, the x-axis, and (a) the ordinates at x = — a, and x = ; (b) the ordinates at x = 0, x = a. 8. Find the area inclosed by the axis of x, and the curve y = x — ^. 9. Find the entire area of the curve y^ = a'x^ — x*. 10. Find the area included between the curve y'^ (a^ _ x^) = a^^ and its asymptote x = a. 11. Find the entire area contained between the curve y'^ (a^ — x^) = a* and its asymptotes x = fl, x = — a. 12. Find the area included by the curve x^y'' (x^ — a') = a" and its asymp- tote X = a. 13. Find the area of the loop of the curve a'y'^ = x* (6 + x). 14. Find the total area bounded by the curve a^t/ + bV = a^b^x^. 15. Find the volume of the solid generated by the revolution about the X-axis of : (a) !/2 = x' — x^ between the ordinates x = 1, x = 2 ; (6) (a^ — x^)?/* = a^ between the curve and its asymptotes x = o, x = — a. 16. Find the volume generated by the revolution about either axis of the hypocyoloid x"'" + j/^ = d^. 32.] GEOMETRICAL APPLICATIONS 77 17. Find the volume of the solid generated by the revolution about either axis of the parabola x^ + y'^=za^. 18. Find the volume of the solid generated by the revolution about the X X 2/-axis of that portion of the catenary s^=-(e«+e «) between the lines X = a, X = — a. 19. Find the volume generated by the revolution of the cissoid y^ = — — — about the a;-axis from the origin to x = a. 2a — x 20. Find the volume generated by the revolution of the cissoid in Ex. 19 about its asymptote x =2a. [Reference may be made to the table of in- tegrals in the Appendix.} 31. Find the volume of the frustum of a cone obtained by rotating about the a-axis the line joining the points (—4, 1) and (3, 6). 22. The hyperbola xy = c^ revolves about the axis of y. Show that the volume generated by the infinite branch extended from the vertex (c, c) towards the jz-axis is equal to the volume of the cylinder generated by the revolution of the ordinate at the vertex about the j/-axis. Show that the area which generates the first volume is infinite. 23. Find the volume of the ring generated by the circle x^ + y^ = 25 re- volving about the line x = 7. 24. Show that in the solid generated by the revolution of the rectangular hyperbola x^ — y^ = (fi about the x-axis, the volume of a segment of height a measured from the vertex, is equal to that of a sphere of radius a. 26. Show that the volume generated by the revolution of one semi-undu- lation of the curve j/ = 6 sin - about the x-axis is one half that of the oiroum- scribing cylinder. 26. The figure bounded by a quadrant of a circle of radius a, and the tangents at its extremities, revolves about one of these tangents ; find the volume of the solid thus generated. CHAPTER V RATIONAL FRACTIONS 33. A rational fraction is one in which the numerator and denominator are rational integral functions of the variables. The fraction is proper when the degree of the numerator is lower than that of the denominator. If the degree of the numerator is greater than that of the denominator, division can be carried on until the remainder is of less degree than the denominator. Suppose that N, D are rational integral functions of X, and that the degree of N is greater than that of D. By- division, in which R is of lower degree than D ; and, therefore, In order to integrate the proper fraction — it is often neces- sary to resolve it into partial fractions. It can be shown that any proper rational fraction can be decomposed into partial fractions of the types A B Cx+G Ex + F X — a (x — ay af+px + q (a^ -\- px -\- q)'' in which A, B, C, G, E, F are constants, r, s positive integers, and x^ + px + q is an expression whose factors are imaginary. For the proof of this and the related theory, reference may be 78 33-34.] RATIONAL FH ACTIONS 79 made to works on algebra.* Here nothing more is done than to work some examples in the principal cases that occur in practice, t 34. Case I. When the denominator can be resolved into factors of the first degree, all of which are real and different. Ex. 1. Find rx^-7x^ + 6x-6^^_ J x^ — x^ — &x Ondivision, x^ - 7x^ + 6x - 6^ ^^ ^ ^ 6(2x-l) a^ — K^ — 6x x' — x^ — 6x 6(2x-l) . x(x-3)(x + 2) Put 6(2x-l) ^±^^_^^^ (1) X (x - 3) (x + 2) X X - 3 X + 2 in which A, B, C are constants to be determined. Clearing of fractions, 6 (2x - 1) =^ (x - 3) (x + 2) + Bx (x + 2) + Cx (x - 3). Since this is an identical equation, the coefficients of the same powers of X in each member are equal. On equating the coefficients of like powers of X, it is found that A + B+ C = 0, -A+2B-3C=12, - 6 A = - 6. On solving these equations for A, B, C, there results A = l,B = 2,C=-3. Therefore, after substituting these values in (1), rx*-7x^ + 6x-6^^^r/ ^, 1^ 2 _ 3 N^^ J x^-x^-6x J\ X x-3 x + 2/ = ^x2 + X + logx + 2 log (x - 3) - 3 log (x + 2) = i^(x + 2)+log^fe^. * See Chrystal's Algebra, Parti., Chap. VIII., Arts. 6-8. t A few remarks on the decomposition of rational fractions are given in Note A, Appendix. 80 INTEGRAL CALCULUS [Ch. V. A shorter method for calculating A, B, C, could have been employed in the example just solved. Since 6(2x-l) _A B C K(x-3)(x + 2) x x-S x + 2 is an identity, it is true for any value of x. Clearing of fractions, 6 (2x - 1) = ^ (X - 3)(x + 2)+Bx(x + 2) + Cx(x - 3). On letting the factor x = 0, A = 1 ; on letting the factor x — 3=0, orx = 3, B = 2; and on letting the factor x + 2 = 0, or x = — 2, C = - 3. Ex 2 C_5i? Ex. 12. f '^^'^^ ■ Ex 3 C^Sx+lld^. E^ 13 r2x3-6x^-4x-ll^ Ex. 4. fli^l^)^. Ex.14. p^ + 7x2 + 6x-6^^_ Jx2-3x + 2 J x2 + 2x Ex. 6. f^^. Ex. 15. r_^!4^2__dx. Jx3_a; J x(x—p){x + q) Ex. 6. CM±1)^. Ex.16, f ^^-3x + 3 ^^ J x(x2-l) J (x-l)(x-2) Ex 7 (■ (a-b)xdx j,^ 17 r (X + l^dx , ■ Jx2-(a+ 6)x + a6 Jx2 + 2x-4 Ex 8 r (3x + l)dx Ex 18 C (8=c^-31^°+41a:-6)(fa ' J2x2 + 3x-2' ■ ■ J 12(x*-6x3+llx2-6x) Ex. 9. CLLzl^)^. Ex.19. (-12x^^5)^. J 3x-x8 J X*- 5x2 + 6 Ex. 10. CIMnl)^. Ex. 20. r ^2 Jx2 + x-6 Jx'-7x5 + 14x3 - 8x Ex.11. r(i + ^. Ex.21. C(^-«)(^-6)(^-«)^^, J x-x8 J (x — a)(x — ^)(x (x-7) Suggestion. — Assume the fraction equal to 1 H — h etc.] 34-35.] RATIONAL FRACTIONS 81 35. Case II. When the denominator can be resolved into linear factors, all of which are real and some of which are repeated. Ex. 1. (-6x3-80=^-4^ + 1^ J 3;4 - 2 x^ + a;2 Let 6a:3-8a;^-4x+l ^AB C D X\X - 1)2 X X2 X - 1 (X - 1)2' in which A, B, C, D, are constants to be determined. Clearing of fractions, 6x3-8x2-4x+ Is^x(x-1)2 + B(x-1)2 + Cx2(x-1)+Z»x2. Equating coefficients of like powers, A + C = &, -2A + B-C-i- D = -%, A-2B = ~4:, B=l. On solving these equations it is found that A = — 2, B — 1, C = 8, D = — 5. Therefore, r6xB-8x2-4x + l^^^ n_2 I ^ ^_u J x\x-iy J\ xx2x-l (x-l)V = -21ogx-i+81og(x-l) + - ^ X X —I = log(x-l)% 4x+l X2 X (X - 1) -i)dx Ex. 2. f_£^_ Ex. 7. f^xCx-il J (X + 1)« J (X - 3)» Ex. 3 ri^^lil^. Ex. 8. fj^izl)^ J (X-3)2 Jx3 + 2x2 + OX2(7j; Ex. 4. r2Cx + 22^. Ex. 9. f J (2x + l)2 J(x + a)a Ex 6. rf-^ *^Vx- Ex.10, r J \x + a (x + 6)2/ J 4x2+^_2^^_ X'' — X'^ Ex 6. (■ 2^ Ex.11. r_(2x-5)^. •'(3V5-2-x)3 J(x + 3){,x + l)2 82 INTEGRAL CALCULUS [Ck. V. Ex.12. (■ "^^ Ex.14. f9(2 + 4a:-a=^)fe Ex.13. C-^^ Ex.15. f (^' + ^)^^ - J (a;2-2)-'2 J a; (a; -1)8 36. Case III. When the denominator contains quadratic factors, the linear factors of which are imaginary. This case can be sub- divided into the two following : (a) When all such quadratic factors are different. (&) When one or more of them is repeated. The latter case seldom occurs in practice. For each power, from the first to the jith, of these quadratic factors, a numerator of the form Mx + N, in which M, N are constants, should be assumed. J x^ + ix Ex. 1. Find Assume * =4 + ^+0. a;(x2 +4) X x^ + i Clearing of fractions, 4 = Aix"' + 4) + x{Bx + O). Equating coefficients of like terms, A + B = Q, G = 0, 44 = 4. The solution of these equations gives A = l, B = -l, = 0. Hence, f ^ ^^ = U~ - -^^\dx Jx^ + ix J\x x-^ + ij = log X — ^ log (a;2 + 4) = log- Vx2 + 4 1. Find r -' (x - i Ex. 2. Find \ 5i+i dx. ■ -2)(x2-2x + .3)2 Assume ^i^ + i ^ A Bx+C Dx + E (x-2)(x-^-2x + 3)^ x-2 x2-2x + 3 (x^ - 2 x + 3)2 33-36] BATIONAL FRACTIONS 83 Clearing of fractioiis, a;3+ l = ^(x2 + 2x + 3)2 + (Bx+ 0')(,x - 2)(^x^-2 x+3) + {Dx+JE)(x-2). Equating coefficients of like powers of x, A + B = 0, -4:A+ C-iB=l, \OA + T B~iC+ D = 0, -\2A~6B + TC+E-2D = 0, 9A-6C-2E=1. The solution of these equations is, A = l, B= — 1, C = 1, Z) = 1, E =\. Hence, C ix^+\)dx n \ x-\ x+\ \^^ J (x-2)(x2-2x + 3)2 J\x-2 x2-2x + 3 (x2-2x + 3)V =iog ^-^ . 1 + r 2^ Vx2-2x + 3 2(x2-2x + 3) J(x2-2x + 3)-^ The last integral can be found by means of reduction formulae to be given in the next chapter. Ex.3. fCE+ilii^ Ex.6. (•M±2)^. J X? + X J (X2 + 1)2 Ex.4. C 2X2 + X + 3 j,^,7_ r (2X2 + X + 2MX J(x+l)(x2+l) J (3x + 2)(2x2 + 4x + 4) j,^_g_ r3x^ + 6x3 + x-^ + 2x + 2^^_ ^^^ r(3x^-17x + 33)fc J 3x3 + 6 J x3-6x2 + llx Ex 9 f a:^ - 3x^ - 22x3 + ITx^ - 23x + 20 ^^ J (x + 4)(x2+l) Ex.10. C^+7x^+.13^^_ Ex.11. r3x^ + 3x3 + .30x^ + 17x + 75^^_ J 3 Cx2 + 5)3 Ex.12. f^i^ xs/x^ - a^ J x' 39. Trigonometric substitutions. Although the integration of trigonometric functions is not discussed until the next chapter, it may be stated here that trigonometric substitutions sometimes aid the integration of irrational functions. The following sub- stitutions may be tried : (a) x = a sin for functions that involve Va^ — x', (b) x= a tan 6 for functions that involve Va^ + x', (c) x=a sec for functions that involve s/a^ — o?. Ex. Find (" Va^ - x^ dx. (See Ex. 1, Art. 23.) On assuming xr= asva.6, dx = a cos 6 dS, and (\/a^-3i'dx= a^^cos^ede = ^({T- + cos 2 e^de = «!(« + |sin20)=~(ff + sin«cosff) 2 2 = ifa2sin-i- + xVa-'-xA- 2\ a I 40. Expressions containing fractional powers oi a + bx only. If n is the least common multiple of the denominators of the powers of a + bx, these expressions can be reduced to the form F(x, ^.a + bx). If F(u, v) is a rational function of u, v, then j F(x, ^/a + bx) dx can be changed into a rational form by means of the substitution a -\-bx = «" 86 INTEGRAL CALCULUS [Ch. VI. For then x = ^—^^^, dx = ^^^dz, and hence, Cf{x, Va + bx) dx = - CFf?^^^^, z^'-' dz. Expressions that contain fractional powers of x only, belong to this class. This is apparent on putting a = in a + bx. If n is the least common multiple of the denominators of the exponents of X, the function can be changed into a rational form by the substitution x=z\ Ex. 1. Find ("- '^^ X y/d^ + hx ,2 /y2 9 g On making the substitution a? -^- bx = z^, x = -— , dx = -—dz. b b Hence, \ — , „ ~ ^ 1 12 J x\/a^ + bx -^2 dz o z + a _ 1 jQg Va'-' + bx-a "• Va'^ + 6x + a /• x^ ■' 1 + x^ The L. C. M. of the denominators of the exponents is 6. If x = 3*, then dx = 6 2^ dz, and -dz = 6x*(jx-|x^ + ixi-l)+6 tan-' x^. Ex. 3. Find f J^dx. Ex 4 Find (- 3 + 5(xT^ + x^>?x -'^-1 -^ 15CX + X*) 40-4h] IRRATIONAL FUNCTION'S 87 Ex. 6. f ^^ + ^ ^. Ex. 9. j f^ r(xi-2^i)^ Ex. 10. (^/^±l±lax. j,^ y -1^-7-^^+12^^ ^^ Ex.11. J(3-s)v'(c-3:)2da;. T. . r / 5- , Ex. 12. C — ^^5 Ex. 8. JxVa+6a;(Zx. •'(2x + 3)^ 41. Functions of the form f\x^, {a + bx^y"}xdx, in which m, n are integers. If f(u, v) is a rational function of m, v, these functions can be rationalized by means of the substitution a + 6a^ = z". For then, 2 &a;da;=n2"~'d2, a;^= 5-JZf^^ and the function becomes b 2¥\ b ' ' Ex. 1. Find ( — -^—-. (See Ex. 10, Art. 38.) This belongs to the form above, since xVx'^ — a'- x^Vx^ — a^ On putting x^ — a^ = z'^, xdx = z dz, x^ = z^ + a^, and r (jx _ r (^g •^ xVS^rr^ Jz^ + a^ = ltan-i?- = ltan-i^^^-«^ = l-cos-xg. a a a X s. 2. f — ^^——. (See Ex. 8, Art. 38.) Ex. 3. (~ 'x-s/x^ + a-'' -^ VI -x2 88 INTEGRAL CALCULUS [Ch. VI. xdx T,^ c f ic'dx Ex.4, , C ^'^^ Ex. 5. f- Ex. 6. If /(«, u) is a rational function of «, «, show that \ 'ex + dl can be rationalized by means of the substitution — '^^- = «". cic + d 42. Functions of the form F{x, ^x^ + ax + b)dx, F(u, v) being a rational function of u, v. If the radical be -Vinx'+px+q, it can be written Vm \h^ + —x + —- The given function can be rationalized by assuming that Va;^ + ax+b = z — x, and then changing the variable from x to z. For, squaring and solving for a;, z'-b x = - ' a + 2z whence, z — x= — — — :^^^—, ' a + 2z ' and cZx = 2(5l±i^d.. Therefore, on substitution, /ET/ /TT" ; n;N J oC-nrfz^—b z^+az+b\z^+az+b -, F(x, Var + aa; + 6) da; = 2 I Fl , — ■ '— ] — ■ ^ dz. Ex. f ^^ Assume Vx^ + x + 1 z= z — x. Squaring and solving for x, x = ^ ~ . l + 2« 41-43.] IRRATIONAL FUNCTIONS 89 Hence. ^^^2(.2 + , + i) (1+2 2)2 "^' and Va2 + a; + 1 = a - k = ^^ + ^ + 1. l + 2» On substitution, f '^^ = 2 f -^?_ •' a; Vjc^ + a; + l J ^-1 = log a;-l+Va:2 + x+l a; + 1 + Vk^ + X + 1 43. Functions of the fonn /(a?, V-as^ + aa; + 6) dsc, /(«, v) being a rational function of u, v. If the radical be V^^maf+px+q, it may be written Vm 'V — a;^ + — a; + — • H the factors of — cff' + ax + b are imaginary, V— x^ + ax + b is imaginary. For, if one of the factors is x—a+i/S, the other must be —(x—a—ift), and hence, — (e' + ax + b = —(x — « + il3){x — a— i/3) = -l(x-ay + li^, which is negative, and has an imaginary square root whatever x may be. Only cases in which the factors of — 3^ + ax + b are real will be considered here. Let — ay'+ax + b=(x — a)(/8 — x). Assume V— ay' + ax + b or V(a; — a)(;8 — x) = (x — a) z. Then, squaring, /8 — a; = {x —a)^, ag^ + S 1 +a^ 90 INTEGRAL CALCULUS [Ch. VI. (1 + zy ' and V - a,-2 + aa; + 6 = (a; - a) 2 = i^^ri^. Hence, on making the substitutions, which is rational, and accordingly integrable. Equally well, the substitution, y/{x — a)(/8 — x) = (fi — x)z, might have been made. It follows from this article and the preceding article that, if X is the indicated square root of an expression of the second degree in X, every rational function / (a;, X) is integrable. Ex. Ex. 1. Find (* '^ •^ xV- x^ + J ■6a;-6 Assume V-a;2+6x-6 = V(x-2)(3-a;) = {x- 2)«. From this, on squaring, 3 — x = (x — 2)z2. Hence, x = ^ "*" , and V- x2 + 5 a; - 6 = (x - 2)s : (2=' + 1)2 z 22+1 Therefore, on substitution. a;V-x2 + 5x-6~ J 2 22 + ; = -\/|tan-i V|2 Ex. 2, ^S >'3(x-2) . f ^5 Ex. 3. f- •' (\A- X2~l VI - X2 •/ -, (1 + x2) VI -x2 ' ' " J V2 x2 + 3 j; + 4 «-44.] IRRATIONAL FUNCTIONS 91 Ex. 4 x^vn^^ CV^xT^^^_ Ex. 6. J— ^ Ex. 5. f /^ + ^ . 1. x^x + a "'(2x + 3)Vr«;2_2 2. r — ^ 16. f- (a — 4) (Zx VaT^Vx^^ ■ ■^ (a; + 3)Vx2 + 4a- + 5 3 r .1 /dx 17_ r <2a; -' V^'-^.c- - Px + R ^ (x^ - a;) Vl + 2x- ^ V:,^^ + V^3^ ^ (separate -^ *"*? P'^'^'i^' fr-'=-\ I ax. \ x^ — X tions. j •^ {X — a) Vx — 6 "J; xcZx ^g r (x'^ + 3x+ 5)dx ^ (x + lWx2+ 1 5. (• ^^'^^ . •^ \/-2 ,'.;" + 3 :r' + 1 6, f ^ "' ^ (x + l)Vi2T -^ V- 3;.y-i -6x -2 20. C ^-f- + '-' a: + 3) dx C dx J {X' + 2 'J- - 3) Vl - x'^ 21- j VP+a--ix3(V6-^ + a2x'Hax)2(Z.i;. V-27 + 10x+ 5x2 r V — m^x + inn -. ■^ (mx — n) Vmx + n gg f -^ (a'^ — X'' (?X j;-2)2Va2 - x2 .34 a: — 17x2 23, f a:'gx ^ Va' - x2 10. 9 r V2x-xHV^^^^''+g)+h ^^ y/cfi — x:' rV2x-3(\/2-3x+V^r + 3) , \ ^ -^---dx. c xcl ^ (2x-3)V0-5x-6x2 24. \—~ 11. r£^i^±iig. . ^, -'Vx2 + 2x + 4 25. J — r(x + iw^ ,^, . •^ V3-a; J Vn2 _ x2 12. t'C ^ + iW^^ ^^, ,,„ r c;x 13. j'(x-l)^(x + l)-5(Jx. 37. J Va^-x^ ^^^^ 14. r f?x /• -'(x.+ l)VS2-:n"' ^*- Ja:2Va2_a;2^x. IRRATIONAL FUNCTIONS 99 :^ dx. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. dx ■' .\/t2 a. f , •^ V2 ax — a;2 r dx •' xy/2ax — 3? r :^dx xdx \/2 ax — x:' J V2 ax — x'^ dx. C dx J (x2 + a2)8' J xdx (X* + a*)''' r x^dx J (x3 - a3)2' r dx J (x2 - 2 X + 3)2' INTEGRAL CaLC. 44 § dx (X- i)Ha;+ 1)^ 45. Vl + X + X2 I^ 46 § + x Vxdx dx. 47. a;3 + 6x2+ nx + 6 2 a6x cZx ■- J- •'(a' 2 + 62-x2)V(a2-x2)(x2-62) 48. I V2 ax - x2 vers-i - dx. Jo a 49 C -x* + x3 + 3x + 9 ^^ J X* + 9 X* + 27 x2 + 27 50. f^^±^^x. •^ Vx2 + 2 VVx2+2 Vx2+2 / ., r x2 + 2 X + 1 -dx. -S: y/ix^ + ix + S x2 + X + 1 52. t ■ " -^- -^" -(^x. Vx2 + 2 X + 3 S3. /; x2 - X - 3 (Zx. V-3 + 12x-9x2 64. \ Vlx^ + mx-\- n dx, I negative. dx (x2 + a2)Vx2-a^ 56. .f- (x+1) (^X (2 x2-2 x + i)V3 x2-2 x + 1 rmr - LIBRARY ■^ 16 1937 AGRIC. ECON. & FARM MQT. CHAPTER VII INTEGRATION OF TRIGONOMETRIC AND EXPONEN- TIAL FUNCTIONS Integration by parts and the use of snbstitvitions 'will be found very helpful in obtaining the integrals of trigonometric and exponential functions. 46. I sin" a; dx, | cos" x dx, n being an integer. (a) If n be a positive odd integer, say 2 m + 1, I sin" a; da; = | sm^'"'''^xdx= | sin^"" a; sin a; da; = — I (1 — cos^a;)'"d (cos x). The binomial in the latter form can be expanded, and the inte- gral found term by term. Ex. 1. { s'm^xdx = — I (1 — oos2a;)ti(cosa;) = — cos a; -I- Icos^a; -I- c. Similarly, I cos^""'"' .!• (?.<; = I (1 — sin^a;)'"d(sin.'B)= sina; — — sin'a; + ■■•. Ex. 2. ( oos^X('Ix= \ {l—8'm^xyd(smx)= \ (1— 2 sin2a;-|-sin^x)«Z(sinx) = sin a; — fsin'a;-!- ^ sin^a; -(- c. (b) If n be any positive integer, integrate sin" a; da; by parts. putting u = sin" ~' X, d'O — sin x dx. 100 40.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 101 Then, du= (n — 1) sin""^ x cos x dx, v = — cos x ; and I sia" xdx = — sin""' x cos a; + (n — 1) j sin" "^ a; cos^ x dx = — sin""* 35 cos x+(n — l) i sin""^;!; (1 — sin^a;)da; = — sin""' X cos x+ (n — 1) j sin""^ a; da; — (n—1) I sin" a; da;. By transposing the last term to the first member, combining, and dividing through the equation by n, the result is /. „ , sin""' X cos X , n — 1 C ■ „_« ■, tat sm" a; da; = I sm" ^x dx. ]A\ n n J ■' The result A can be used as a rednction formula. Successive applications of it leads to | da; or | sin a; da; according as n is even or odd. Ex. 3. fsins xdx=- ^^^ '^°°^ ^ + 1 fsin x dx 1 2 = — sin^xcosa; — cos a; + c. 3 3 This result may be compared with that of Ex. 1. On integrating cos" x dx by parts, putting u = cos""' x, dv = cos X dx, the following reduction formula is obtained, /sin a; cos""' a; , n — 1 f „ „ , r-p-i cos" a; da; = 1 I cos""^ x dx. L^J n n J The deduction of B is left as an exercise. (c) Suppose that n is a negative integer. The value of j sin" "''a; da; in A is rsin»-^a;da; = ?H^^^:^^^ + -^ fsin-xda;. J n — 1 n — IJ 102 INTEGRAL CALCULUS [Ch. VII. 46- On changing n into n + 2 this becomes /. . , sin"+' X cos X , n + 2 ^ ■ „ , , , r-r^ sin" xax= '— I sin"+^ x dx. \C\ n + 1 n + lJ "- -' This can be used as a formula of reduction when n is a nega- tive integer. Ex. 4. I -r~" = i (sin x)-^dx = — ^ cos rc(sin x)-^ + i (sin x) "i dx ^__cosx _|_ fesc^.^^^ 2 sm^ X ^ 1 X = — cot X CSC a; + log tan - + c. 2 ^ ^ 2 Similarly, on solving for | cos"~^a;da; in B, and then changing n into 71 + 2, there results, /.,„„!. ™j™ sina;cos"+^a; , n + 2 /* „+2 j r-n-i cos" a; aa; = — I cos + a; dx. \D\ n + 1 n + lJ '- -' This is a formula of reduction for I cos".«da; when n is negative. It is advisable to remember the method of deriving A, B, C, D, so that they may be readily obtained when necessary. Ex. 5. (a) icos'xdx, (6) jsin^x^x, (c) jsin'xtix. Ex. 6. (a) |sln*xdx, (6) jsinSxdx, (c) isinSxdx. Ex. 7. (a) lcos*xdx, (6) j cos^xdx, (c) j cos'xdx. Ex. 8. (a) ij~^, (h) f^, (c) f-^, (d) f^, J sm* X J cos^ X J cos* x J cos^ x ^ cos°x J sm*x J sin^x Ex. 9. Show that PsinZ-^dx = 5-LllAj::Ll(2nLllI) .£. Jo 2.4.6..-2m 2 Ex. 10. Show that ("^ sin^^+ix (fa = ^ ■4-6 ■■■2m — Jo 3.5.7^^^2m + l 48.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 103 47. Algebraic transfonnations. The trigonometric integral I sin" a; da; can be put into an algebraic form. For, if then cos xdx = dz, and dx = -^ = ^-^ cos a; VI — z^ and hence, j sin" a; da; = ( ^" • J J VI — z' The second member has a form which has been discussed in Chapter VI. If the substitution cos x = z be made, j sin" a; da; = — \ (1 — z^) ~^~dz. This form can be integrated by methods already explained. These substitutions may also be employed in the case of j cos" x dx. Ex. 1. i SLTi^xdx. (Compare with Ex. 3, Art. 46.) On putting = — \ sin^ X cos a; — I cos x + c. Ex. 2. Solve Exs. 2, 4, 6 (a), 7 (a), 8 (6) of Art. 46 by algebraic sub- stitution. 48. I sec" X dx, I cosec" x dx, 1 1 r (a) Since seca; = , and coseca; = , isec^xdx and /cos.t' sm.i; J cosec"a;da; can be reduced to the forms considered in Arts. 46, 47. 104 INTEGRAL CALCULUS [Ch. VIl. Ex.1, (a) jsec^xcfe, (b) j ooseo^ . dx, W | J^, ^ Jh^S?^- [See Exs. 8 (c), 8 (g), 8 {d), 6 (a), (Art. 46).] Ex. 2. Find T — , assuming x = a tan 0. (See Art. 39, and Ex. 2, J (x2 + a2)' Art. 45.) (b) If 71 is an even positive integer, another method may be employed. Since sec^ x = 1 + tan^ x, and d (tan x) = sec^ a; da;, I sec" a; da; = | sec"~^a; sec^a;da; = I (1 + tan^a;) ^ d(tana;). The binomial under the sign of integration can be expanded in »i — 2 a finite number of terms since n is even, and accordingly — ^ — is an integer. In a similar way it can be shown that n-2 I cosec" .c (Ix = — I (1 -I- cot^ a;) ^ d (cot a;). Ex. 3. ("sec^ xdx= (" ( 1 + tan^ a;)2(7(tan x) ; = tan X + f tan' x + J tan^ x + c. (Compare Ex. 8 (e), Art. 46.) Ex. 4. (a) Isec'j'dx; (6) iooseo^xdx; (e) I coseo^ x dx ; {d) I sec* | (Zx. [Compare the results with those of Exs. 8 (c), 8 (/), 8 (gi), Art. 46.] (c) If n is any positive integer greater than 2, the method of integration by parts can be used. On putting sec""^ x = u, sec^ x dx = dv, it follows that (/k =(« — 2)sec""^a; tan a;da;, 'y = tana;. 48.] TRIGONOMETBW AND EXPONENTIAL FUNCTIONS 105 Hence, | sec" xdx= i sec"~^ x sec^ x dx = tan X sec""^ a; — (w — 2) | sec"~^ a; tan^ a; dx. On substituting sec^ a; — 1 for tan^ x in the last term, and solv- ing for I sec" x dx, there is obtained /„ , tan X sec"~^ x , n —2 C „_, , r \-\ sec" xdx = 1 I sec" ' x dx. [A] ft — 1 n — IJ This formula of reduction leads to | sec x dx when n is odd, and to I dx when n is even. Similarly, integration by parts will give /„ , cot X cosec"~^ a; , ft — 2 Z' „_, ■, run cosec" a; da; = 1 I cosec" ^ x dx, [B] ft — 1 n — IJ which, on repeated applications, leads to j cosec x dx or to I dx, according as n is odd or even. Ex. 5. (a) ( sec' x dx ; (6) t sec* x dx ; (c) | cosec' x dx ; (d) I cosec^ xdx; (e) | sec' | x dx. [Compare the results with those of Exs. 8 (6), 8 (d), 4, 8 (a), Art. 46.] Ex.6, (a) (seG*xdx; (6) (cosec*xdx. [Compare Ex. 4 (a), (6).] (d) Transformation to an algebraic form. fly If ta.nx = z, x=ta,n-^z, dx = - -, sec^x = l+z'; 1 +z^ 11-2 and hence, j sec" xdx= j {1+z^ ^ dz. Also, if sec x = z, 106 INTEGRAL CALCULUS [Ch. VII. 48- dz it follows that dx=- z-^z^ - 1 and hence, I sec" xax= I — -^^:= »-i dz. In like maimer, the substitutions z = cot a;, 2 = cosec x, will re- duce I cosec" X- dx to an algebraic form. Ex. 7. Solve Exs. 3 (a), 4 (a), 4 (6), 1 {d) above, by algebraic substitution. 49. I tan^xdx, | coV^xdx. (a) Let n be a positive integer. Then, | tan" xdx= | tan"~^ a; tan^ x dx = I tan"^^ X (sec^ a; — 1) (Za; = I tan" ^ a; d (tan a;) — j tan""^a; da; = ^^'^" y - ftan" - a; da;. [A] n — 1 J This reduction formula leads to | dx or to I tan x dx, accord- ing as II is even or odd. Ex. 1. I tan' X (?x = j tan x (seo^ x — l)dx— \ tan xd (tan a;) — i tan x dx = J tan^ X — log sec x -j- c. In like manner, I cot" X dx = I cot"~^ X cot^ a; r/a; = j cot"^'' x (cosec^ a; — 1) dx ^_ cot"-'a; _ rco^.„-.3.(^3.^ |-j3-] another reduction formula. 50.] TRIGONOMETRIC AND EXPONENTIAL FUNVTIONH 107 (b) If n is a negative integer, say — m, j tan" xdx= j cot"* x dx, and | cot" xdx= j tan" x dx. Hence, this case reduces to the preceding. Ex. 2. (a) ^tanixdx, (6) (tanPxdx, (c) ftane^dx, (t?) (cot^xdx. (e) |cofix&, (/) Jcof'xdx. (c) Transformation to an algebraic form. If tana; = s;, dx= • dz and hence, j tan" xdx= j — Again, if sec x = z, dx + z^ dz z-yjz^ — 1 11- 1 tan" a; da; = | ^^ ^ — dz. Similarly, | cot" a; do; can be changed into algebraic forms by the substitutions, z = cot X, and z = cosec x. Ex. 3. Solve Ex. 1, 2 (a), 2 ( ™ — 1 whence, -d = , -is = — ; m+n m+n Hence, ( sin"' X cos" OR dx - ' X cos n + l. m + n + — — ^rsin™-2a5Cos"a3da3. [A] m + nJ In a similar way, by connecting | sin" a; cos" a; (7* with the other integrals mentioned in the rule, the following reduction formulae are obtained : rsin"* as cos" x dx = -^ '"+ixcos» + * J. «, « J sm"' + ^a[;cos sin"* as cos" x ax JW + 1 m + n + i r^ij,™ + 2 ^ g^g„ ^ ^^ |-Bj m + 1 /». /.ak"— 1 ^ + ^~^ C sin™ X COS" - 2 as dx. [C] m-\- nJ I sin™ X cos" X dx = - si" * ""^ '^ J sin + ^ + "'^ + ^ fsin™ X COS" + 2 X dx. [D] W.+ 1 J In each of the four integrals with which | sin" a; cos" a; da; may be connected by the rule, m or n is increased or diminished by 2. The numerical values of m, n will indicate the one of the four which is simpler than | sin'" w cos" a; dx, and with which it may preferably be connected. A succession of steps like A, B, C, D, may be necessary. 51.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 111 In solving the problems of this and the following article, it may happen that the results will not agree in form with those given in the answers. An agreement can be made by using the trigonometric relations, sin^ x + cos^ a; = 1, sec^ x=l + tan^ x, etc. Any apparent difference in results will be due to a difference in the methods of working the examples. Ex.1, j sin^ X cos^ X (to;. Assume I siu^xcos^xc^x = ^sinxcos^x + -B I cos^xdx. Differentiating, sin^ x cos^ x = A cos'' x — H A sin^ x oos^ x + S oos^ x, whence, dividing by cos^x, sin^x = ^(1 — sin^x) — 3 ^sin^x + B. Equating coefficients of like terms, -4^ = 1, A + B = 0. On solving these equations, A = — ^, -B = J. Hence, | sin^ x cos^ xdx = — J sin x cos^ x + ^ j oos^ x dx ■ from this, by Art. 46, = — \ sin x cos^ x + J (sin x cos x + x) + c. Equally well, I sin'' x cos^ x dx might have been connected with \ sin-'xdx. Also, 1 — cos^x miglit have been substituted for sin^x, or 1 — sin^x for cos^x, and the integi-al found by the method of Art. 46. Ex.2. Csin^xcos^xdx. Ex.4. f^^cZx. J J sm-* X Ex.3, f '^ ■ Ex.5. r£24^c?x. Jsin*xcos2x Jsva^x Ex. 6. Solve some of these examples by reducing them to an algebraic form, as described in Art. 50 (c). (b) The results A, B, C, D can be used as formulae of reduc- tion. It is necessary only to substitute in them the proper values for m and n. It is not necessary to memorize these formulae. The student should make himself familiar with the process of deriving these formulae, so that he can readily obtain them when 112^ INTEGRAL CALCULUS [Ch. VII. 51- I'equired.* The formulge A, B, C, D of Art. 46 are special cases of A, B, C, D above. This will be apparent on putting m and n in turn equal to zero in the latter formulae. Moreover, | tan" x dx, I cot" X dx, discussed in Art. 49, may be put in the forms I sin" a; COS"" a; da;, | cos" a; sin-" a; da;, and solved by the methods of this article. For the sake of practice in making the substitu- tions a few examples may be solved by means of the formulae. Ex. 7. I sin6 X cos*xdx. By A, Csins ^ cos* xdx = - sinSxcosSa; _^ 6_ C^^^ ^ ^^^4 ^ ^^ . by A, ("sin* x cos* xdx = - s'"'' a: cos^ a: _^ 3 r^^^ ^ cos* x dx ; ■J 8 o*/ by A, f sin2 x cos* xdx = - ^'"^°°^^^ + i fcos* x dx ; by C, fees* xdx = sinacosag _^ ^C^^^2 ^ ^^ . byC, Jcos=x(?x = 5iE^2S2 + iJdx=?HI^2s^+.^ + c. The combination of the results gives I sin^ X cos* xdx = — ^ sin^ x cos ^ x — ^ sin^ x cos^ x — -^-^ sin x cos^ x + xiT^i^^oos'^ + i|^sinxcosx+ ■s^x + c. The formula? might have been applied in other orders, for example CACAA, CCAAA, etc. Ex. 8. Solve Exs. 2, .3, 4, 5 by means of the reduction formulae. * Another method of deriving these formulae, namely, by integrating by parts, is given in Note C, Appendix. Other formulae of reduction can be obtained by connecting 1 sin" X cos" X dx with j sin"'-2xcos»+2x(?x, | sin'»+2xcos"-'ixdx in the manner described above. The formulae for these cases may be derived as an exercise. (See Edwards, Integral Calculus, Art. 83.) 52.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 113 52. J tan"* as sec" as d!a?, | cot*" as cosec" as das. (a) Reduction to the form j sin-"a;cos^a!da;. This may be done by the substitutions , sin a; 1 , cos a; 1 tan X = , sec x = , cot x = -: — , cosec x= - cos a; cos a; sma; sma; and the integration can then be performed by one of the methods of the last two articles. (b) Reduction to an algebraic form. n-2 2 dz. If tan x = z, I tan" a; sec" a; da; = | «"'(1 + z^) This is almost immediately integrable if m is a positive even integer. In this case, | tan"' x sec" x dx can be reduced to inte- grals of the form | tanPa;d(tana;). (See Ex. 1, below.) tan" a; sec" a; da; = |z"~^(«^ — 1) ^ dz. This is almost immediately integrable if m is a positive odd integer. In this case, J tan"a;sec"a;da; can be reduced to integrals of the form j sec'a;d(seca;). (See Ex. 1, below.) The form j cot" a; cosec"a;da; may be treated in a similar manner. Ex. 1. ( tan^ X sec* xdx= \ tan^ x sec^ x sec'' x dx = I tan^xCtan^x + l)d(t&nx) = J tanfi X + I tan* a; + c. Or, j tan^ x sec* xdx = \ tan^ a; sec^ x sec x tan x dx = i (sec2 X — 1) sec^ x d (sec x) = J sec« X — J sec* x + c. 114 INTEGRAL CALCULUS [Ch. VII. 52- Ex. 2. I Ex. 5. i cot^xcosec^Kdz. Ex.3. (ts.\3xs&c^xdx. Ex.6. \ X.a.n' x &%(fi x dx. Ex, 4. I oot^ a; coseo* X (?x. Ex.7. I tan^ x sec* x (ix. Ex. 8. Solve some of these examples by using algebraic transformations. 53. Use of multiple angles. When m and n are positive and one of them is odd, the first method of integration shown in Art. 50 can be employed in the case of | sin" x cos" x dx. When m and n are positive and both even, the use of multiple angles will aid the process of integration. The trigonometric substitutions that can be employed for this purpose are : sm X cos X = ^ sm 2 x, cos^.'B = 1(1 + cos 2 x), sin^ a; = ^(1 — cos 2 x). Ex. 1. j sin2 X cos^ xdx = \\ sm? 2xdx = 1 1 (1 — cos 4 x) dx = Jx — ^jSin4x + c. (Compare Ex. 1, Art. 51.) Ex.2, fsin^xdx. (See Ex. 6 (a), Art. 46.) Ex.3. Coos^x^x. (See Ex. 7 (a), Art. 46.) Ex.4, (sm'^xdx. (See Ex. 6(6), Art. 46.) Ex. 5. ("cosOxcZx. (See Ex. 7(6), Art. 46.) Ex.6, fsiniarcos^xdx. (See Ex. 2, Art. 51.) ' Ex. 7. j sin* X cos* x dx. 55.] rRIQONOMETRIC AND EXPONENTIAL FUNCTIONS 115 54 f —^ On denoting the integral by I, and dividing the numerator and denominator by cos^ .)', J _ r se c- X dx _ / * d (tan a;) On substituting u for tana;, integrating, and replacing u by tan x, there is obtained, 7=i-tan-^^*^M^ ab fct r dx C dx ■ J a + b cos as' J a + 6 sin as* Since cos x = cos^ - — sin^ '-, and cos^ - + sin^ n ~ ''■' Z Z Z Zi f tzx _ r dx ^^^ J a + 6 cos x J J^Q^2 X ^ g- j^2 ^\ ^ J / g2 .« _ g.^, ^\ On dividing numerator and denominator in the second member by cos^- and reducing. / dx ^ 1 r a + 6 cos X a — hJ. sec' ? dx 2 a — 6 d ( tan - ) "-^-'tan^^ + ^i^ 2 a— & /fZ;^ C dz -, -„, or I 5, accord- z^ + c- J V- — & ing as a is greater than 6, or less than h. Hence, if „ > ,, f-^^- = -^= tan- fV^tau |\ INTEGRAL CALC. 9 116 INTEGUAL CALCULUS [Ch. VII. bo- if a < 6, f- J a dx y/b + a + Vfe — a tan | + 6cosa;- VF^T^"" VsT^ - V6^^ tan ^ ;log tanh' tan- On introducing the half -angle as before, and dividing numerator and denominator by cos^ -, as in the case just considered, it will be found that, r dx _ r J a -\-h sin x^ J dx a| cos^- + sin^- 1 + 26 sin- cos- V 2 2y 2 2 =/: iX J sec* - dx a + 2b tan - + a tan^ - ^ Li = -/■ a J I r'^2 + a)+-^ This is in the form I —^ ^, or i — ^, according as a is greater %/ % -f— c %/ % ~~ c or less than h. Hence, if a > &, f- dx tan" a tan -+ b if a < 6, f- + 6sina; Va^ - 6' ( VcF+ 1 , 2 da; a tan - + 6 — V6^ — a^ log- a + 6sin.r V6^-a^ a tan^ + 6 + VF:r^' V6^ - a' coth" a tan — + 6 2 V6' - a^ 56.] TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS 117 In working the examples it is preferable to follow the method employed above, and not to use the results that have just been found as formulae for substitution. Ex. 1. Ex.2. Ex. 3 C^ Ex.4, r J 3 — 2 sin a; J dx (■ ^ Ex.8, f '- J2 + 3sin2a; Ji + 5 r_^ — j,^ g r J 5 + acosx J 4 5 — 3 COS* dx 5 cos 2 X dx 5 + 3COSX J4 + 5sin2x Ex. 7. When a = 6 in this article, find the integrals. 56. I c"* sin nx doc, ie"'" cos nx dx. On integrating e"" sin nx dx by parts, first taking e'" dx for dv, and then taking sinw^da; for dv, there are obtained /' . , e"" sin nx n f a^ j /^ n e" sm nx dx = \ e'" cos nx dx, (1) a a J /' „» • J e'" cos nx , a C „r j /n\ e" sm ?ia; da; = \-- \ e'" cos na; aa;. (2) n nJ The integral in the second members of (1), (2) can be elimi- nated by multiplying the members of (1) by - and the members n of (2) by -, and adding the results. When this is done it will be found that /' , • J e"" (a sin nx — n cos nx) /o\ e"' sm nx dx = — ^ ^- (3) a^ + m^ Similarly, on integrating e"" cos nx dx by parts, first taking e'"dx for dv, and then taking cos najcte for dv, and eliminating j e"" sin ma; da; which has thus been introduced, there will be obtained /' , e'"(w sin Ma; + a cos na;) ,.. ^ cos nx dx = — !^ — -— i- (4) or + n^ 118 lyjEGRAL CALCULUS [Ch. VII. The result (4) can be obtained by eliminating i e"'' sm.nxdx from (1) and (2). It can be deduced also by substituting the result (3) in (1) or in (2). It is, however, preferable to deduce it directly by integrating by' parts. As in Art. o"! the student is advised to work the examples by the method followed above, and not to use (3) and (4) as formulse for substitution. Ex. 1. (e'^si-a.xdx. Ex. 4. J^^^da;. Ex. 2. (e'cosxdx. Ex. 5. i^^dx. Ex. 3. \ e'^' cosZ 'X dx. Ex. 6. \ e^ cos?- x dx. 57. rslnHixcos/fvTf?,*-, \ci)%inxc«%nxdx, i^inmxsmnxdx. Since, by trigonometry, sin mx cos nx = -^ sin (m + ") * + 1 siu ("i — n)^, C ■ 7 cos (m, -\- ri) X cos (in — ri) x I sm inx cos nx ax = — i ' — i ^ i — J 2(wi + «) 2{m — n) In a similar way it can be shown that, /, sin (m + n) x sin (m — n) x cos mx cos nx dx = '- — i } ^— , 2(m + M) 2(m-n) f sin mx sin )tx dx = — sin (m 4- )() x sin (m — n) x 2(7». + li) 'J(m — ii) E.K. 1. I COS 3 X sin 5 .'■ f?..-. Ex. 4. ( cos3x cos Jxdx. Ex.2, j cos4a:cos7.rda;. Ex.5, ("cos ^ asin ^xtfo. Ex.3, j sin 5 x sin 6 X dx. Ex.6. ("sinyV^sin J^xdx. CHAPTER VIII SUCCESSIVE INTEGRATION. MULTIPLE INTEGRALS 58. Successive integration. It has been seen in the differential calculus that successive differentiation with respect to x is some- times required in the case of functions of the form u=f(x); and that successive differentiation with respect to both a; and y may be required in the case of functions of the- form u=f(a:,y). On the other hand, the reverse process called successive inte- gration is sometimes necessary. This chapter will be concerned with describing the notation that is used in "multiple integral tion," as it is often termed ; and it will show, by examples, how successive integration is introduced and conducted. Arts. 61, 62, 63, contain applications of multiple integration to the measure- ment of areas in rectangular coordinates, and of volumes in rec- tangular and polar coordinates. Plane areas in polar coordinates and curvilinear surfaces will be found by means of multiple integration in Arts. 67, 75. 59. Successive integration with respect to a single independent variable. Suppose that f^(x)=jf(x)dx, (1) f2(^)=fM^)dx, (2) 119 120 INTEGRAL CALCULUS [Ch. VIII. and fa (x) = j /2 (x) dx. (3) Since /a («) = jE/i («)] dee, it follows from (1) that /j (»)=(" Cf{x) dx dx ; (4) and since /^{x) = | {/2(a;)} da;, it follows from (4) that f^{x)= i \ | | /(«) tia; da; ^ dx. (5) The second member of (4) is usually written in a contracted form, namely, I \f{x)dxdx, or | |/(a;)da;^, (6) in which d«? means (da;)^, and not d (a;^. Similarly, the second member of (5) is usually written rrr/(a;)da;da;da;, or f f r/(a;) da;'. (7) Integral (6) is called a double integral, and integral (7) is called a triple integral. In general, if an integral is evaluated by means of two or more successive integrations, it is called a inultiple in- tegral. If limits are assigned for each successive integration, the integral is definite ; if limits are not assigned, it is indefinite. Ex. 1. Determine the curve for every point of wliicli the second differ- ential coefficient of the ordinate with respect to the abscissa is 8. The given condition is expressed by the equation (1) ^ = 8. \dxl dx wlience, dl —\ = Sdx, This may be written — - — = 8 ; 59.] SUCCESSIVE INTEGRATION 121 Integrating, (2) ^=8a; + c, whence, dy =(?>x + o)dx. Integrating again, (3) y = 4:X^+ cx + k. Tliis is tlie equation of any parabola that has its axis parallel to the y-axis and drawn upwards, and its latus-rectum equal to 4. All such parabolas will be obtained by giving all possible values to c and k, the arbitrary constants of integration. Two further conditions will serve to make c and k definite. For instance, suppose that the tangent to the .parabola at the point whose abscissa is 2, is parallel to the x-axis ; and also that the parabola passes through the point (3, 5). By the former condition, ^ = when a; = 2 ; dx andhence, by (2), = 8-2 + 0, that is, c=— 16. Equation (3) then becomes y = 4 a^ _ ig a; + A. Also, since the parabola passes through the point (3, 5), 5 = 4.3^-16.3 + *; whence, k = 17. Therefore the equation of the particular parabola that satisfies the three conditions above is 2/ = 4a!2-16x + 17. The given relation (1) might have been written in the differential form, cPy = ?< dx^, and y expressed in the form of a multiple integral, namely, ^=j'j'8da;2; whence on integrating, = 1 (8 a; + c) da; = 4 x^ + ca; + A:. The former solution is better because it shows all the steps more clearly. Ex. 2. If s represents distance measured along a straight line, and t time, — is the velocity of a body that moves in the straight line, and — is its dt dfi acceleration or rate of change of velocity. In the case of a body falling in a vacuum in the neighborhood of the earth's surface, the acceleration or rate 122 INTEGRAL CALCULUS [Ch. VIII. of increase in tlie velocity is constant and equal to about 32.2 feet>-per-second per second. The number 32.2 in this connection is denoted by the symbol g. Let it be required to determine s from the known relation, '(I) This may be written — =- — = g. (1) = ^'^'' Using the differential form* d and integrating, (2) — = gt + c, in which c is an arbitrary constant of integration. Writing the latter equation in the differential form, ds = (gt + c) dt, and integrating, (3) s = J gfi + ct + k, in which k is another arbitrary constant of integration. In order that the constants c, k may have definite values, two further conditions are required. For instance : (a) .Suppose that the body falls from i-est, and that the distance is measured from the starting point. ds In this case, s = 0, and — = 0, when t = 0. dt Hence, substituting in (2), = + c, that is, c = ; and, substituting in (3), = + + A, that is, A: = 0. Therefore, the distance through which a body falls in a vacuum on starting from rest is J fft^, in which g is about 32,2 and t is the duration of fall in seconds. (6) Suppose that the body has an initial velocity of 8 feet per second, a" (3 that the distance is measured from a point 12 feet above the starting point. By the last condition, s = 12 when { = ; and hence, by (3) , 12 = + + k, whence k = 12. 59-60.] SUCCESSIVE INTEGRATION 123 By the other condition, — = 8 wlien t = ; dt and hence, by (2), 8 = + c, that is, c = 8. Therefore, under these conditions, s = igfi + 8t + 12. The linown relation (1) might have been written in the differential form, d^s = gdfl ; from this, s = ( I gdfi ; whence, on integration, s = \ {gt + c) dt = {gP + ct + k. Ex. 3. Evaluate ( C T^x^ (dx)^ The integrations are made in order from right to left. Thus, if / denote the integral, = 41 fxl dx = 8 I dx = 16. Ex. 4. Evaluate C C Cx^dxy. Ex. 5. Evaluate T f^ ("'«" (.^^Y- (Compare Exs. 4, 5, with Ex. 3.) Ex. 6. Determine all the curves for which -^ = 0. Ex. 7. Find the curve at each of whose points the second derivative of the ordinate with respect to the abscissa is four times the abscissa, and which passes through the origin and the point (2, 4) . Ex.8. Find C ('' ( "r*(^dry. Ex.9, p (^ (" sine (dey. 60. Successive integration with respect to two or more indepen- dent variables. In this article the notation commonly used in this kind of integration will be described ; and, in preparation for the next article, a few examples will be given so that the student may become familiar with the notation. Suppose that /^(x, y, z) =jf{x, y, z) dz, (1) 124 INTEGRAL CALCULUS [Ch. VIII. the integration indicated in the second member being performed as if X, y were constants. (It will be remembered that if a;, y, •■•, are independent variables, differentiation of F (x, y, •■■,) with respect to one of the variables, say x, is performed as if the others were constants.) Then, suppose that f^{x, y, z) =jMx, y, z) dy, (2) the integration now being performed as if a;, z were constants. Again, suppose that f^(x, y, z) = J/sCa;, y, z) dx, (3) the integration being performed as if y, z were constants. Equar tion (2), by virtue of equation (1), can be put in the form fi{x,y,z)=j\yf{x,y,z)dz dy; (4) and equation (3), by virtue of (4), can be written, f^(x, y, z) = J I J j'f{x, y, z) dz dy |- dx. (5) The bracketing in the second member of (5) indicates that the differential coefl&cient, f{x, y, z), is integrated with respect to z ; that the result of this integration is then integrated with respect to y ; and that, finally, the result of the last integration is inte- grated with respect to x. In the notation usually adopted, the second member of (5) is abbreviated by removing the brackets, and the order of the variables with respect to which the integra- tions are made, is indicated by the order of the respective differ- entials of the variables beginning at the right and going toivard the left. Thus, the abbreviated form of the second member of (5) is jjj /(a;, y, z) dx dy dz. (6) 60,] SUCCJBSSIVE INTEGliATION 125 This is a triple integral. Similarly, the double integral in the second member of (4) is generally written, Jjf{x,y,z)dydz. As to the integration signs, the first on the right is taken with the first differential on the right, which is dz in (6) above, the second sign from the right is taken with the second differential from the right, the third sign from the right is taken with the third differential from the right, and so on. It is well to note this usage, because attention must be paid to it when limits of integration are assigned to x, y, z* In some of the examples below, and often in practical problems, the limits for one variable are functions of one or more of the other variables. Ei. 1. Evaluate | ( j xy^dxdydz. If /denote the integral, 1= C( \zYxy'^dxdy = Z^ Cxy^dxdy Ex. 2. Evaluate I ( xy^ dx dy. Jo Jiz Ex.3. CCCxy'dzdydx. Ex.4. C C Cxy^dzdxdy. (Compare Exs. 3, 4 with Ex. 1.) — /*b riot , Ex. 5. £'£ (w + 2 v^dv dw. E^- 7- j^ y V^^^^ dtds. Ex.6, t I r^siaffdedr. Ex.8, i ( \ x^yzdxdydz. Jo Jo Jo Jo Jiy * The notation described above is not universally adopted, but it is the one most frequently used. 126 INTEGRAL CALCULUS [Cii. vin. 61. Application of successive integration to the measurement of areas : rectangular coordinates. In this and the two following articles, problems are solved which show applications of succes- sive integration. In some of the examples there may not be any special advantage in resorting to double integration, for the reason -that a single integration may suffice. They are, however, given to the student for the purpose of making him familiar with an instrument for solution which may sometimes be the only one possible. It will be found that the elements in the summations which follow are infinitesimals of a higher order than those which have been met with heretofore. Fig. 81. Ex. 1. Find the area included between the parabolas whose equations are 3 !/2 = 25 x, and bx^ = 9y. The parabolas are VOP, WOP. Their points of intersection, 0, P, are (0, 0), (3, 5). Taking any point Q within the area as a vertex, construct 61.] SUCCESSIVE INTEGRATION 127 a rectangle whose sides are parallel to the axes of coordinates and are equal to Ax, Ay. Produce the sides which are parallel to the x-axis until they meet the curves in L, M, G, B, thus forming the strip LGBM, and produce ML to meet the y-axis in B. On Lil construct the rectangle HM, giving it a width Ay. ■ax=o/ / Area of the rectangle H3I = limit > Ax Ay. Both y and Ay remain unchanged throughout this summation. Now, BL=^, and BM = 3-\^^. Hence, area ^ilf= t dx At^ (1) As Ay approaches zero, the rectangle HM approaches coincidence with the infinitesimal strip GM, and, in the limit, HM coincides with GM. Also, the area OLPMO is the limit of the sum of all the strips similar to LGBM lying between and P, when Ay is made to approach zero as a limit. Therefore, i i=py _ area OLPMO = limitAjio V^ (s-y/^ - ^\Ay = 5. If the linear unit is an inch, the answer is in square inches. On substi- tuting for ( S-y- — '~ ) in (3) its value as shown by (1) and (2), there is obtained, areaOiPilfO= Tj fj ^°dx]dy •4 -£y. dy dx. (4) 128 INTEGRAL CALCULUS [Ch. VIII. The latter is the customary abbreviated form which indicates that the first integration is made with respect to x between the limits -J^ and 3-v/| for X, and that the result obtained thereby is to be integrated with respect to y between the limits and 5. The element of area in (4), namely dydx, is an infinitesimal of the second order. Another way of performing the double summation required in adding up all of the elements of area like dy dz, may be described as follows. Sum all of these elements that are in the vertical strip ST, and then sum all of the vertical strips in OLPilO. In the first summation, x and dx do not change, and the upper and lower limits of y are 5a/|, 2^ respectively ; in the second summation, the limits are the values of x at and P, namely and 3. This double summation is indicated by the double integral 0. ^dxdy. 9 This, on evaluation, gives an area of 5 square units as before. The area OLPMO might have been expressed in terms of single integrals. For OLPMO = OLPN- OMPN Jo ^Z Jo 9 = 10 - 5 = 5. Ex. 2. Solve Exs. 7-11, Art. 29, by this method. 62. Application of successive integration to the measurement of volumes: rectangular coordinates. If the equation of a surface is given in the form f{x, y, z) = 0, the volume can usually be determined by means of three successive integrations. In the particular case of solids of revolution, the volume can be found by a single integration. This was shown in Art. 30. In Art. 61 the element of area was dydx, the area of an infinitesimal rectangle each side of which was an infini- tesimal. In the case now to be considered, the element of volume will be the volume, dxdi/dz, of an infinitesimal parallelepiped each of whose infinitesimal edges is parallel to one of the axes of coordinates. This will be illustrated in Ex. 1. •^1-62.] SUCCBSSIVE INTEGRATION 129 Ex. 1. Find the volume of the ellipsoid whose equation is ^ + 2^4.^-1 a2 -^ 6-2 + c2 ~ ^• Let 0-ABC be one eighth of the ellipsoid whose volume is required. Then OA = a, OB = 6, OC = c. Take IL an infinitesimal distance dx on OX, and through /, L pass the planes HIJ, KLM perpendicular to OX. Take EF of an infinitesimal length dy on ML, and complete the infinitesimal rectangle EFGfD. Through the lines DE, GF pass planes parallel to the plane ZOX which intersect the curvilinear surface SJMK in the infinitesi- mal arcs B V, ST. Through a point D' on DM, DB' having an infinitesimal length ds, pass a plane parallel to the plane XO Y. The infinitesimal paral- lelepiped B'F, whose volume is dxdydz, will be taken for the element of volume. The solid 0-ACB is the limit of the sum of parallelepipeds of this kind. This limit will now be determined. First, the volume of the vertical rectangular column RF will be found by adding together all the infinitesimal parallelepipeds such as D'F which are included between DEFG and B8TV. 130 INTEGRAL CALCULUS [Cii. VIII. Second, the volume of the slice HIJMLIC will be found by adding together all the infinitesimal rectangular columns like RF which are erected between IL and J)L Third, the volume of 0-ABC will be found by adding together all the .infinitesimal slices lilie HIJMLK that lie between OCB and A. In the addition of the infinitesimal parallelepipeds from DEFQ to BSTV, z alone varies, and it varies from zero to DS. Vol. BF -- t=0 dydx. (1) In the addition of the vertical columns from IL to MJ, y alone varies, and it varies from zero to U. ( !i=Ijr z = DR .. Yo\. HIJMLK =\ f ^dz 1 dy \dx. (2) In the addition of the slices between OGB and A, x varies from zero to OA. Vol. 0-ABC -- x=OA I !/ = rj " 1=0 Lf=o Z=DR' dz z = dy > dx. Writing this in tht- usual manner. x=OA y = IJ z=DR Vol. 0-ABC = f j f dxdy dz. (3) (4) x = y = z = ll If the coordinates of B are x, y, s, it follows from the equation of the surface that DB = z, :c^ji-=^-y± a^ 62 At the point J, z = 0; and hence, \ /J.2 Also, OA = a. Therefore (4) becomes Vol. 0-ABC Jo Jo Jo dx dy dz. 02-03.] SUCCESSIVE INTEGRATION 131 On making the integrations in their proper order, it is found that Vol. 0-ABC = iirabc. Hence, the volume of the whole ellipsoid = ^ wabc. Note 1. The volume of an infinitesimal parallelopiped is an infinitesimal of the third order, the volume of a vertical column is an infinitesimal of the second order, and the volume of a slice is an infinitesimal of the first order. Note 2. Equally well, the planes hounding an infinitesimal slice might have been taken perpendicular to either OZ or Y. Note .3. On putting a = 6 = c, the volume of a sphere of radius a is foimd to be I ira'. Ex. 2. Find the volume of the ellipsoid given in Ex. 1 : (o) by taking the infinitesimal slice at right angles to OT; (6) by taking it at right angles to OZ. Ex. 3. Determine the volume of a sphere of radius a by the method of this article. Ex. 4. Find the volume bounded by the hyperbolic paraboloid z = ?■'', the aiy-plane and the planes x = a, x = A, y = b, y = B. " Ex. 5. Find the volume of the wedge cut from the cylinder x^ + y^ = a- by the plane z = 0, and the part of the plane z = x tan a for which z is positive. Ex. 6. Find the entire volume bounded by the surface Ex. 7. The center of a sphere of radius a is on the surface of a right 'Under the radius of whose base is cylinder intercepted by the sphere. cylinder the radius of whose base is -• Find the volume of the part of the 63. Further application of successive integration to the measure- ment of volumes: polar coordinates. The illustration in this article is given, because the use of polar coordinates in dealing with solids is often advantageous. It will be necessary to employ these coordinates in solving some of the problems in Arts. 77, 79. Ex. 1. To find the volume of a sphere of radius a by means of polar coordinates. Let a point on the surface of the sphere be the pole, the tan- integkal calc. — 10 132 INTEGRAL CALCULUS [Ch. VIII. gent plane at be the a;!/-plane, and the diameter through he the ^-axis. Take any point P within the sphere, and let its coordinates be denoted by r, e, 4),—r being its distance OP from 0, B the angle that OP makes with the 2-axis, and ^ the angle that the projection of OP on the xy-plane makes with the K-axis. Draw PM perpendicular to OZ. Produce OP an infinitesimal distance dr, and revolve OP through an infinitesimal angle d9 in the plane ZOP. The point P thus traces an infinitesimal arc of length rd9. Complete the infinitesimal rectangle PQ that has the sides dr, rde just described. Now let the angle (t> be increased by an infinitesimal amount d4>. Then P and the rectangle move through a distance equal to MP dtp, that is r sin 6 df. The rectangle will thus have generated a parallelepiped whose edges are dr, r dd, r sin B d, and whose volume therefore is r^ sin 8 dB dtfi dr.* The volume of the sphere is the limit of the sum of such parallelepipeds. Hence, * Tliis is not absolutely coiTeot, for the opposite edges of the solid gener- ated differ in length by infinitesimals of higher orders. By fundamental theorems in the differential and integral calculus these differences will not affect the limit of the sum of the infinitesimal solids. See Art. 67 and Note D. 63. 1 SUCCESSIVE INTEGRATION 133 Volume of sphere = C \ C C r'^dr \d(p >■ sin fl de = I, ^^0 I- r = = 1 I I r^smeded that is, area AOB = i (^r^d9, by the definition of a definite integral. The element of area for polar coordinates is thus ^ r'dO. Ex. 1. Find the area of the sector of the logarithmic spiral whose equation is r = e"*, between the radii vectores for which « = a, S = /3. Area POQ = i (^r^de . POQ = i Jj Ja 4a ri, rj being the bounding radii vectores. FiQ. 85. Ex. 2. Find the area of one loop of the lemnisoate r'^ = a^ cos 2 5. The area of one half the loop, OMA = h frW, between proper Umits for 6, which must be determined. 138 INTEGRAL CALCULUS [Ch. IX. The initial and final positions of the radius vector are OA and OL the tan- gent to the arc OM at 0. For r = 0, the equation of the curve gives = a^ cos 2 9 ; and hence, 29 = ± or e-. ■■^T The positive sign indicates the position of OL, and the negative sign that of ON. Hence, area OMA -~€ r^dfi 2 Jo Hence, area of loop OMARO = — . 2 Ex. 3. Find the area of a sector of the spiral of Archimedes, )• = a6, between d = a, S = p. a Ex. 4. Find the area of the part of the parabola r = a sec^ - intercepted between the curve and the latus rectum. 1 1 n Ex. 5. Find the area of the cardioid >■- = a^ cos -. 2 Ex. 6. Find the area of the loop of the folium of Descartes, x^ + y^ — 3 axy = 0. 3 g sin 9 cos 6 _ cos'fl + sinSfl.' and ( Hint . Change to polar coordinates, thus obtaining r = then cliange the variable $ by putting z = tan 6.) Ex. 7. Show that the area bounded by any two radii vectores of the hyperbolic spiral rO = a, is proportional to the difference between the lengths of these radii. Ex. 8. Show that the area of a loop of the curve r^ = a^ cos n0 is — ■ n 66-67.] FURTHER GEOMETRICAL APPLICATIONS 139 67. Areas of curves when polar coordinates are used : by double integration. The areas of curves whose equations are given in polar coordinates can bB found by double integration, in a man- ner analogous to that used in Art. 61. Example 1 below will serve to make the method plain. Successive integration in two variables, polar coordinates, will also be required in Arts. 77, 79. Ex. 1. Find by double integration the area of the circle whose equation is )• = 2 n cos d. Let OLAN be the given circle, O being the pole and OA the diameter 2 a. Within the circle take any point P with coordinates (r, 9). Draw OP and produce it a distance A)- to S. Revolve the line OPS about through an angle A9 to the position OQR. Then area PQRS = ^ [ (r + Ar)2 - r^] A9 = r Ar A9 + J (Ar)2 ^0. Fig. 37. Produce OP, OQ to meet the circle in M, G. The area of the sector MOG will be found by adding all the elements PQBS therein, and the area of the semicircle OLMA will be calculated by adding all the sectors like MOG that it contains. Area MOG = ^PQRS :limitAr = o/ ^ r=OX r = = limits,, ^0 7^ j-ArAe by a fundamental theorem in the calculus,* r=OM and hence, area MOG = | rdrAff, r = * See Note D, Appendix. 140 INTEGRAL CALCULUS [Ch. IX. in which the integration is performed with respect to r. Hence, e=TOX r=OM = TOX r= OM area OLMA = C \ irdr\dd Jo Jo 2acoB0 rdedr _ira-' 2 ' Hence, the area of the circle is ira?. The area of OMA can also be obtained by finding the area of the circular strip L9 whose arcs are distant r, r + dr from O, and then adding all of the It similar concentric circular strips from to A. The angle LOG- = cos"^ will be found that 2a area OMA ■ r2a /'cc Jo Jo rdrde = ^^ as before. 2 Ex. 2. Find by double integration the area of the circle of radius a, the pole being at the center ; (1) by adding equiangular sectors ; (2) by adding concentric circular strips. 68. Areas in Cartesian coordinates with oblique axes. In this case the method of finding the area is similar to that in Art. 61. Let the axes be inclined at an angle . Pig gg The whole area is the limit of the sum of all parallelo- grams that are constructed within the perimeter of the figure when Aa; and Ay approach zero. Hence, the area is the value of the double integral I I sin o) dx dy, that is, sin w j j da; dy, 67-69.] FURTHER GEOMETRICAL APPLICATIONS 141 between, the proper limits for x and y. If the element of area be an infinitesimal strip parallel to the y-a,xis, as in Art. 27, the area is the value of the integral sin (o i ydx between the proper limits for x. 69. Integration after a change of variable. Integration when the variables are expressed in terms of another variable. The function under, the sign of integration may assume a much simpler -form on changing the variables. Examples 1, 3 illustrate this. Some- times, also, the ordinary variables are expressed in terms of an- other variable. Examples 2, 4 illustrate this. When a change is made in the variable or variables, the corresponding changes should be made in the limits. The work of returning to the original variables, in order to substitute the original limits, will thus be avoided. These remarks are also applicable to practical examples in other articles. Ex. 1. Determine the area of the circle whose equation is x^ + y'^ = a^. (Compare Ex. 3, Art. 27.) Area of circle = 4 I ydx. Jo Put X = a cos ff. Then y = a sin d, and dx = — asia8 d$. Also, B = — when x = 0, and 8 = when x = a. 2 Making these substitutions in the integral above, r" area of circle = — 4 I a'' sin^ d dd = _4a^j-;(L^i^)c : ira^, Ex. 2. Find the area between the x-axis and the complete arch of the cycloid whose equations are x = a (S — sin 9), y = a(l — cos 6). Area= t ydx. Jo l42 INTEGRAL CALCULUS [Ch. IX. When a; = 0, 6 = 0; and when x = 2 wa, 6 = 2ir. Also, dx = a(l — cos B) dff. K these values for y, dx, and the limits, be substituted in the integral above, it becomes area = a^ (" ^''(l - cos 9)2 de Jo oC^^f-, n o , l + cos2e\ ,„ :o2l M - 2 cos g + ^ jde = Sira^. That Is, the area is three times that of the generating circle. x^ w2 Ex. 3. Find the area of the ellipse -^ + 1^ = 1. (Compare Ex. 3, Art. 27.) (Hint : put X = a cos tj>, then y = b sin (/>.) Ex. 4. Find the volume of the solid generated by the revolution of a complete arch of the cycloid of Ex. 2 about the x-axis. 70. Measurement of the volumes of solids by means of infinitely- thin cross-sections. In Art. 30 the volume of a solid of revolution was determined by finding the volume of an infinitely thin slice of the solid, the slice being taken at right angles to the axis of the figure, and the sum of the volumes of all such slices being then found. This method can be extended to other figures besides figures of revolution. Some convenient line is chosen, and an infinitesimally thin slice of the solid is taken at right angles to this line. If the area of a face of the thin slice can be expressed in terms of its distance from some point on the line, the volume of the slice can be expressed in terms of this distance ; and from this, the sum of the volumes of all the slices can be found. Por example, let the chosen line be taken for the axis of x, and sup- pose that the area of a face of a thin slice at right angles to this line is /(«). Let the thickness of the slice be Ax. The volume is (as in Art. 30) the limit of the sum of an infinite number of infinitesimal cylinders whose volumes are of the form f{x) Ax. That is, 69-70.] FURTHER GEOMETRICAL APPLICATIONS 143 volume of the solid = limit ^^j,, 2 f{x) Aa;, =Jf{x)dx, in which the limits of integration are determined from the figure. Ex. 1. Determine by this method the volume of the ellipsoid ^ + Vl + ^=l (The student is advised to make a figure. ) At a distance x from the center cut out a very thin slice at right angles to the a;-axis, aud let its thickness be dx. The face of this cylindrical slice will be au ellipse whose semiaxes are These values are deduced from the equation of the ellipsoid. The area of this ellipse = ?i6cjl ]• Hence, the volume of the slice = irbc ( 1 — — j Aa; ; and therefore, volume of ellipsoid = ir6c| [1 ^dx = f irabc. Ex. 2. Pind the volume of a sphere of radius a by this method. Ex. 3. Find the volume of the torus generated by revolving about the K-axis the circle x^ + (y — 6)^ = a^, in which 6 > a Ex. 4. Find the volume of a pyramid or a cone having a base B and a height h. Ex. 8. Find the volume of a right conoid with a circular base and alti- tude h, the radius of the base being a. Ex. 6. A rectangle moves from a fixed point, one side varying as the distance from this point, and the other as the square of this distance. At the distance of 2 feet, the rectangle becomes a square of 3 feet. What is the volume then generated ? Ex. 7. Given a right cylinder of altitude ft, and radius of base a. Through a diameter of the upper base two planes are passed touching the lower base on opposite sides. Find the volume included between the planes. Ex. 8. Find the volume of the elliptic paraboloid 2x = ^-{ — cut ofl by the plane x = h. P H 144 INTEGRAL CALCULUS [Ch. IX. 71. Lengths of curves: rectangular coordinates. To find the length of a curve is equivalent to finding the straight line that has the same length as the curve. For this reason the measure- ment of its length is usually called "the rectification of the curve." * The deduction here made of the integration formulae for the length of a curve depends upon the definition that integra- tion is a process of summation. The equation of a given curve is f(x,y)=0; it is required to find the length s of an arc AB, A being the point (xi, y^, B be- ing the point {x^, yi). On the curve take any two points P, Q whose coordinates are x, y, and X + Ax, y + Ay. Draw the chord PQ and make the construction indicated in the figure. FiQ. 39. The chord PQ = y/{Axy + {Ayf =^R1- (1) (2) As Q approaches infinitely near to P, that is, when Ax ap- proaches zero, the chord PQ approaches coincidence with the arc PQ. It is shown in the differential calculus that if Ax is an infinitesimal of the first order, the difference between the * In 1659 Wallis (see footnote, Art. 27) published a tract in which he showed a method by whioli curves could be rectified, and in 1660 one of his pupils, William Keil, found the length of an arc of the semi-cubical parabola : ay'. Tliis is tlie first curve that was rectified. Before this it had been generally supposed that no curve could be measured by a mathematical proc- ess. The second curve whose length was found is the cycloid. Its rectifi- cation was effected by Sir Christopher Wren (1632-1723) and published in 1673. This was before the development of the calculus by Leibniz and Newton. 71.] FURTHER GEOMETRICAL APPLICATIONS 145 arc and its chord is an infinitesimal of at least the third order ; that is, arc PQ = chord PQ + ig, (3) in which % is an infinitesimal of the third order when A* is an infinitesimal of the first order. Therefore, s = S(arcs PQ) = limit,,.„^(Vl + (kI J^^ + ') = limit,..„2Vl+(!lJ^-' x = x. by a fundamental theorem.* As Aa; approaches zero, — ^ in gen- eral approaches a definite limiting value, namely, — ^. Therefore, by the definition of a definite integral. In applying this formula it will be necessary to express -i/l-f-(-^] in terms of x before integration is attempted. Instead of being put in the form (2), equation (1) may be given the form, chord PQ =yjl + f^YAy. By the same reasoning as above, it can then be shown that 'C^HSf"- s in which \/l + | — | must be expre^ed in terms of y before inte- gration is performed. Formula (4) or formula (5) will be used, * See Note D, Appendix. 146 INTEGRAL CALCULUS [Ch. IX. according as it is more convenient to take x ox y for th.e inde- pendent variable. If As denotes the length of the arc PQ, it follows from (3) that Aa; * VAa;/ Aa; Aa; » \J^xj Aa; Therefore, by the differential calculus, dx V \dx) whence, ds=y\l+(-^\dx. Similarly, ds =-y 1 + ( j- ) ^V- (dx In order to recall formulae (4) and (5) immediately whenever they may happen to be required, the student need only remember the construction of the triangle PQR, and let its sides become infinitesimal. Ex. 1. Find the length of the circle whose equation is oi? + y'^ = a^. Let AB be the first quadrantal arc of the circle. In this case, ^ — _5. dx y Hence, arc AB = (""-v/l + f^ Vdx = (""-v/l + -<^« ^0 ' \dxl Jo ' y^ = rJ^jiiidx = a r ^^ • Jo > 2/2 Jo ^^2 _ a.2 = arsin-i5]'' = : L aj„ JO 2' Therefore, the perimeter of the circle (= iAB) = 2jra. Ex. 2. Find the length of the arc of the parabola from the vertex to the point (xu !/i). Find the length of the arc from the vertex to the end of the latus rectum. 71-72.] FURTHER GEOMETRICAL APPLICATIONS 147 Ex. 3. Find the length of the arc of the semicubical parabola ay'^ = x^ from the origin to the point (xi, y{). Also to the point for which x = ha. Ex. 4. Find the length of the arc of the catenary y = " (e" + e"") from the vertex to the point (xi, j/i). Also to the point for which x = a. Ex. 5. Find the length of the arc of the cycloid from the point at which 9 = ffo to the point at which fl = ffj. Also find the length of a complete arch of the curve. Ex. 6. Find the entire length of the hypocycloid a;* + ;/' = a'. Ex. 7. Show that in the ellipse X = a sin 0, y = h cos 0, being the complement of the eccentric angle of the point (a;, y), the arc s measured from the extremity of the minor axis is s = a\ Vl — e^sin^dxZi^, Jo and that the entire length of the ellipse is Pl 4 a I Vl - e2 sin2 d0, Jo Jo in which e is the eccentricity.* > 72. Lengths of curves : polar coordinates. The equation of a curve is /(r, ff) = 0, and the length s of the arc AB is required, A being the point (rj, ^i), and B the point (rj, $2). On the eurv« take any two points P, Q, whose coordinates are r, 0, r + Ar, & + ^6. Draw OF, OQ, and the chord PQ. About as a center, and with a radius equal to OP, describe the arc PR which inter- sects OQ in B, and draw PR^ at right angles to OQ. Then the angle POQ = ^e, RQ = Ar, and arc Fig. 40. PR = rA(9. * This integral, which is known as "the elliptic integral of the second kind," cannot be expressed, in a finite form, in terms of the ordinary func- tions of mathematics. See Ex. 8, Art. 83. INTEGKAI, CALC. 11 14» INTEGRAL CALCULUS [C" I^'- It is shown in the differential calculus that when A6 is an infini- tesimal of the first order, PBi = arc PB — %, an infinitesimal of the third order ; QRi = QB + i2, an infinitesimal of the second order ; chord PQ — arc PQ — i'^, an infinitesimal of the third order. In the right-angled triangle PRiQ,, chord Pq = VpW+R^- Hence, when \0 approaches zero. arc pq- i\ = ^{PB-i,y+{Bq + i,y, or, arc PQ = V(rAe - kf + {i^r 4- hf + i\ (1) ^J^ +(^- ^ + 2 i,^!^ ^^^^' ^e + i'„ (2) 6) Ai9 '(A^)^ (J which differs by an infinitesimal of at least the first order from V-+f!-:T- Va^ Therefore, s = SPQ = l"^i*Ae^o^V'^ +('^Y • A^, when A6 approaches zero, — approaches the definite limiting dr ^^ value — . Therefore, by the definition of a definite integral. ■=JN-^+(i)^'^«- («) It will be necessary to express -\Ar' +( — ) in terms of 6 before integration is made. ^ ■' Similarly, on removing Ar from the radical sign in (1), it can be shown that 72-73.] FURTHER GEOMETRICAL APPLICATIONS 149 in which -Wl + r^l — ) must be expressed, in terms of r before inte- gration is made. Formula (3) or (4) will be used according as it is more convenient to take d or r for the independent variable. In order to recall these formulae immediately it is only necessary for the student to remember the construction of the figure PBQ, and to suppose that its sides are infinitesimal. Ex. 1. Find the length of the circumference of the circle whose equa- tion is »• = a. Here ^ = 0. = at de = 2 7ro. Ex. 2. Find the length of the circle of which the equation is »• = 2 a sin e. Ex. 3. Find the entire length of the cardioid, r = a(l — cosfl). Ex. 4. Find the arc of the spiral of Archimedes, r = ad, between the points (n, Si), (ra, «2). Ex. 5. Find the length of the hyperbolic spiral, rd = a, from (n, Si) to Ex. 6. Find the length of the logarithmic spiral, r — e"^, from (1, 0') to (n, Si). Ex. 7. Find the length of the arc of the cissoid r = 2 a tan S sin 8 from the cusp (« = 0) to 9 = -• Ex. 8. Find the length of the arc of the parabola r = a sec^ - from 8 = to = 81; also, from 9 = - - to e = -. 2 2 73. The intrinsic equation of a curve. Let PQ be the arc of a given curve, and let s denote its length. Suppose a point starts at P and moves along the curve towards Q. At the instant 150 INTEGRAL CALCULUS [Ch. IX. of starting the point moves in the direction of the tangent PT^. In passing over the arc PQ the direction of motion changes at every instant, until at Q the point is moving in the direction of the tangent QT^. The total change in direc- tion, as the point moves from P to Q, is measured by the angle <^ between the two tangents. It will be found that a rela- tion exists between the distance s through which the point has moved, and the angle <^ by which the direction of its motion has changed. This relation between s and <^ is called the intrinsic f^t equation of the cwve. The form of this equation depends only on the nature of the curve, and the choice of the initial point P. On the other hand, the form of the equation of a curve in other systems of coordinates, for ex- ample the rectangular and polar, depends upon points and lines that are independent of the curve. Hence the term "intrinsic." To find the intrinsic equation of a curve given in rectangular or polar coordinates, (1) Determine the length of arc s measured from some con- venient starting point up to a variable point on the curve. (2) Find the angle between the tangents at the initial and the terminal points. (3) Eliminate the rectangular or polar variables from the equa- tions thus found. Fig. 41. Ex. 1. Find the intrinsic equation of tlie catenary X X y =^(e' + e"'). If the vertex of the curve be taken as starting point, (1) s = I (e° - e "). [Ex. 4, Art. 71.] 73.] FURTHER GEOMETRICAL APPLICATIONS 151 Also, since the tangent at the vertex is parallel with the ataxia, X X (2) tan,(,=^ = i(«"-e"°). The elimination of x from (1) and (2) gives the required equation between s and 4>, viz., s = a tan 0. It is easy to extend this result and show that s = a [tan (4> + (j>i) — tan 0i] is the mtrinsic equation of the catenary when any point A is chosen for the Initial point. The angle ' is the angle that the tangent at the point (r, 6) malies with the polar axis. But 2 0' = 9 + TT. Hence, e = 2 sec (j> + a log tan( _ + - ) EXAMPLES. 1. Find the intrinsic equation of a circle with radius r. 2. Find the intrinsic equation of the cardioid r = a(l — cos 9),. the arc being measured from the polar origin. 3. Find the intrinsic equation of the cycloid x = a(e — sine),\ y = a(l — cos 9), / 152 INTEGRAL CALCULUS [Ch. IX. (1) the origin being the initial point, (2) the vertex being the initial point. 4. Find the intrinsic equation of the parabola y^ = ipx, (1) the vertex being the initial point, (2) the extremity of the latus rectum being the initial point. 5. Find the intrinsic equation of the semioubical parabola 3 ay^ = 2 a;', taking the origin for initial point. 6. Find the intrinsic equation of the curve y = a log sec -, taking the origin for the initial point. 7. Find the intrinsic equation of the logarithmic spiral r = ae'^. 8. Find the intrinsic equation of the tractrix a: = Vc" - yi' + c log " + ^"^ ~ ^^ y taking the point (0, c) as the initial point. 9. Find the intrinsic equation of the hypocycloid x^ + y^ = a', taking any one of the cusps as initial point. 74. Areas of surfaces of solids of revolution. Suppose that the surface is generated by the revolution about the a^axis of the arc AB of the curve whose equation is y =f{x) ; and let the coor- dinates of the points A, B, be a;,, y-^, and x^, y^, respectively. Take (Fig. 42) any two points on the curve, say P, Q, whose coordinates are x, y, and x + Ax, y + Ay. Draw the chord PQ 73-74.] FURTHER GEOMETRICAL APPLICATIONS 163 <^-x- PlG. 48. and the ordinates BP, SQ, and suppose that LM is an ordinate which is not less, and that TN is an ordinate which is not greater than any ordi- nate that can be drawn from the avoPNMQ to the !B-axis. (In TTig. 43, LM coincides with SQ, and TN coincides with EP.) Through N, M, — draw lines PiQi, P-iQij parallel to the a5-axis and equal in length to the arc PNMQ. On the revolution of the arc AB about OX, each point in AB describes a circle with its ordinate as radius. The surface generated by arc PNMQ ^ 2 wLM X arc PQ, and ^2 irTN xasc PQ. When Aa; is an infinitesimal of the first order, arc PQ = chord PQ + ij, an infinitesimal of at least the third order; LM= BP+i, an infinitesimal of at least the first order; TN= BP—i', an infinitesimal of at least the first order. Hence, since the chord PQ =\/l + ( ) ^^! i* follows that 9 \AxJ AaJ + i < surface generated by arc PQ ^2; Therefore, »+')(V^^^^+'')- limitA. = 0^2 ,r fy -A Ml+T^jAa; + %) 154 INTEGRAL CALCULUS [Ch. IX. ^ surface generated by AB < limiW=o^2 nfy + i\ (Jl +f^'- ^« + 4^ By a fundamental theorem * the least and the greatest expres- sions in this inequality are each equal to limitA(c=o V2 Tryyjl + f-^\ ■ Aw. \Aa;y Hence, surface generated by AB = limitAa!=o^2iry^l +(t^) ^a;, When Aa; approaches zero, — ^ takes a definite limiting value, namely — • Therefore, by the definition of a definite integral, area of surface = ^^' 2 wyyjl + (^\ ^ doc. (1) It is necessary to express the function under the sign of inte- gration in terms of x before integration is performed. If aRI)^^^ be used for the length of the chord, there will result. area of surface = f '•'' 2 ir j/ a/i + f -- ") ^ dy . (2) •'yi ' \dyj Formula (1) or formula (2) is taken according as it is more convenient to choose x or y for the independent variable. The surface generated by the revolution of AB about the y-axis is given by the formulae, * See Note D, Appendix. 74.3 FXJBTBEB GEOMETRICAL APPLICATIONS 165 8nrface=J^y^2.xVl + (g)^c?j,, (3) «urface = j-;^2..Vl + (gfc*-. (4) The student is advised to deduce these formulee for himself. The expressions under the sign of integration in formulEe (1), (2) may both be written 2 wy ds hy Art. 71, and those in formulse (3), (4) may both be written 2wxds. In order to recall immedi- ately a formula for the area of a surface of revolution, it is only necessary to remember that the area traced out by an infinitesi- mal arc in its revolution about any line is equal to the product of the length of the infinitesimal arc by the length of the circle which is described by a point on the arc. Ex. 1. Find the surface generated by the revolution of a semicircle of radius a about its diameter. Let the diameter be the avaxis, and the origin be at the center ; the equation of the curve will be a;2 + 2/2 = a2. Surface generated by ABA' about X-axis -'i•r»^Rif- "". ■+(!)' Hence, a;2 -f. j,2 aj y' y^ surface : dx = 4 ira^. Ex. 2. Find the surface of the prolate spheroid obtained by revolving about the avaxis the ellipse 6"V + a'y^ = aP-V^. Ex. 3. Find the surface generated by revolving about the x-axis the parabola 2/^ = 4 ax. Show that the curved surface of the figure generated by the arc between the vertex and the latus rectum is 1.219 times the area of its base. 156 INTEGRAL CALCULUS [Ch. IX. Ex. 4. Find the surface generated by revolving about the j/-axis the (i — ^\ c" + e " ) from x = to x = a. Ex. 5. Find the entire surface generated by revolving about the a;-axis the hypooycloid x^ + y^ = a*. Ex. 6. A quadrant of a circle of radius a revolves about the tangent at one extremity. Find the area of the curved surface generated. Ex. 7. The oardioid »• = «(! + cos d) revolves about the initial line. Find the area- of the surface generated. 75. Areas of surfaces whose equations have the form z=fix,y). Areas of surfaces of revolution vs^ere considered in the last article. A more general case will now be discussed. In the explanation of the following method for measuring the area of a surface, reference will be made to these two geometrical propositions : (a) The area of the orthogonal projection of a plane area upon a second plane is equal to the area of the portion projected multi- plied by the cosine of the angle between the planes. (See C. Smith, SoJkl Geometry, Art. 31.) (h) If the equation of a surface be in the form z =f(x, y), the cosine of the angle between the x(/-plane and the tangent plane at , any point (x, y, z) of the surface is {•KSJKDT (See G. Smith, Solid Geometry, Arts. 206, 26.) Let z = fix, y) be the equation of the surface BFGMALB whose area is required. Take two points P, Q, whose coordinates are x, y, z, x + 6.x, y + Ay, z + Ax, respectively. Through P and Q pass planes parallel to the ?/«-plane and let them intersect the surface in the arcs ML, M^L^ Also pass planes through P, Q, parallel to the zx-plane. The curvilinear figure PQ is thus formed. The projection of the surface PQ on the xy-plane is the rectangle P^Qi whose area is Aa; Ay. When Aa;, Ay approach zero, the point Q comes infinitely close to P; and the curvilinear sur- ^4-75.] FURTHER GEOMETRICAL APPLICATION, S 1; face PQ, which is then infinitesimal, approaches coincidence with that portion of the tangent plane at P, which also has P^Q^ for its projection on the a;y-plane. The area of P^Q^ also becomes dxd,i. Fig, 46. Now let Q be infinitely near to P. If y is the angle between the a;i/-plane and the tangent plane at P, it follows from (a) and the remarks which have just been made, that area PiQi = area PQ • cos y. Hence, area PQ = area Pj Qi ■ sec y = dx dy sec y. Therefore, by (6), area PQ=-yJl + ('—)'+ (^dx dy. The summation of all the infinitesimal surfaces PQ in the strip LMMiLi gives area of strip L y=.SL , D* Ax. 158 INTEGRAL CALCULUS [Ch. IX. The summation of all the strips like LMi in the surface BFOMALB gives area of surface BFOMALB =J f f -^1 + (^' + f-£\ dyUx- 1 = ^y = ~^ or, abbreviating in the usual way. The limits y = SL, x = OA can be determined from the equar tion of the surface. It is necessary to express the function under the signs of integration in terms of x and y. It may happen that a more convenient form of the equation of the surface is either x=f(y, z), or y =/(z, x). The area of the surface will then be the value of either one or the other of the double integrals //>RiNiT*-. //V'HIT-eT--' between the proper limits of integration. In some cases, there are two surfaces each of which intercepts a portion of the other. In finding the area of the intercepted portion of one of the surfaces, it is necessary to obtain the partial derivatives that are required in the formulae of integration, from the equation of the surface whose partial area is being sought. This is illustrated iu Ex. 2. Ex. 1. Find the surface of the sphere whose equation is x2 + 2/2 + 0= = a\ Let 0-ASC (Fig. 45) he one eighth of the spliere. In this case, dx z dy z' andhenee X + l^-lX' ^ [^-lY = X +^1 ^Vl =^ = - \dxl \dyl z-^ z^ z^ a^ ;2 _ j;2 _ „2 75.] FURTHER GEOMETRICAL APPLICATIONS 159 Therefore, area of surface 45 C= f (" Ji + (9Ey^ (dzy^. Jo Jo ■\/a' — x^ — y^ 2 Jo 2 Hence, area of all the surface of the sphere is iwa^. (Compare Ex. 1, Art. 74.) Ex. 2. The center of a sphere, whose radius is a, is on the surface of a right cylinder the radius of whose base is J a. Find the surface of the cylin- der intercepted by the sphere. On taking the origin at the center of the Pig. 46. sphere, an element of the cylinder for the a-axis and a diameter of a right section of the cylinder for the x-axis, the equation of the sphere will be x' + y^ + z^^ a^, and the equation of the cylinder, x^ + y^ = ax. 1(30 INTEGRAL (JALCULUS [Ch. IX. The area of the strip CP will first be found, and then the strips in the cylindrical surface APBOOA will be summed. The element of surface in the strip CP is dxdz. Hence, Cylindrical surface intercepted = iAPBOCA x = Q z=0 Since the surface required is on the cylinder, the partial derivatives must be derived from the second of the equations above. Hence, dy _ a — ^x dy__Q_ dx 2y ' dz Also, CP' = z'' = a' — (x2 + y'), since P is on the sphere, and hence, =a' — ax, since P is on the cylinder. Moreover, OA — a. Therefore, the cylindrical surface intercepted -nr"['-{^)']'-- But on the cylinder, y' = ax — x'. Hence, the intercepted cylindrical surface ^'■'-" dxdz =2.rf ■''> •'» Vax - a;2 = 2ar^^^^d. = 2arJ~Ux -'" Vax - x2 -'o >x Ex. 3. In the preceding example, find the surface of the sphere inter- cepted by the cylinder. Ex. 4. Find the area of the portion of the surface of the sphere x2 + 2,2 + z2 = 2 aJ/ lying within the paraboloid y = Ax' + Bz'. 76. Mean values. The mean value of n quantities is the nth part of their sum. Let (a), cl>(a + h), (a + 2h), ■■■, cj>(a + (n -l)hy, <^(a) + <^(q+ h) + {a)h + (a+n — lh)h 6 — a Now suppose that x takes all the possible values, infinite in number, that are in the interval between a and b. Then, n is infinite, 7t is infinitesimal, and the number of terms in the last numerator is infinite. The sum of all these terms, by Art. 4, is expressed by (x) dx. £ Hence, the mean value of all the values that a continuous function, (x), can take in the interval & — a for x is J>^'«) dx b — a This is usually called the mean value of the function . If p be the mean density, it follows from the definition that, {pdA I' \ka^ feo2 '27r' 79.] APPLICATIONS TO MECHANICS 171 Ex. 8. Find the center of mass of the thin plate described in Ex. 1. ( pxdv i kxyxdA Here, t pdv M -'Ij: " ' x^ydxdy M Aa y = Aa- Similarly, Ex. 3. Find the mass-center for a thin hemispherical shell, radius a, whose density at each point of the surface varies as the distance y from the plane of the rim. Let the hemisphere be described by revolving the semicircle of radius a and center about the y-axis Y, which is at right angles to the diameter, the point being taken for the origin of coordinates. Let P, whose ooordi- Y nates are x, y, be any point on the semicircle, and draw PM, PN at right angles to the axes of x and y respectively. Join OP, and denote / jf - the angle NOP by 0. At the point P, J/ = a cos $ ; also at P, p cc y, that is, p —ha cos 6, in which k is some constant. The infinitesimal arc of length ds at P describes a zone about Y whose area is given by dS = 'i.i!NPds. or, since ds = ade, = 2 ;ra sin 9 • a (?9. The symmetry of the figure shows that Also, {pyAS 2 tt/ot* P cos2 « sin e de y'^^ 2irka^ij^ cosffamede Hence, the center of mass is at the point (0, | a). 172 INTEGRAL CALCULUS [Ch. X. Ex. 4. Find the center of mass of a right circular cone of height h, which is generated by the revolution of the line y = ax about the a;-axis, when the density of each infinitely thin cross-section varies as its distance from the vertex. Symmetry shows that the center of mass is in the a:-axis. Suppose that a very thin plate BS is taken which cuts the axis of the cone at right angles at C at a distance x from the vertex. The radius CB of the cross-section = ax. The density of tliis thin plate, p = kx. The volume of the thin plate, dV= w CR^ dx = ira%^ dx. Hence, (pxdF kTva?-\ x^dx \ (>dV kita?- \ x^ dx Ex. 5. Find the mean density of the cone described in Ex. 4. Ex. 6. Find the mass-center of the surface of the cone in Ex. 4. Ex. T. Find the mass-center of the cone generated in Ex. 4, and the mass- center of its convex surface when the density is uniform. Ex. 8. Find the mass-center of a quadrantal arc of the hypocycloid 4 i- 1 Ex. 9. Find the mass-center of the convex surface of a hemisphere of radius 10. Ex. 10. The quadrant of a circle of radius a revolves about the tangent at one extremity ; prove that the distance of the mass-center of the generated curved surface from the vertex is .870 a. Ex. 11. Find the mass-center of the semicircle of y? -^ ■tp- = cC- on the right of the y-axis. Ex. 12. Find the mass, the mean density, and the mass-center of the semicircle in Ex. 11 when the density varies as the distance from the diameter. 79-80.] APPLICATIONS TO MECHANICS 173 Ex. 13. Find the mass-center of a circular sector of angle 2 o, taking the origin at the center, and the x-axis along the bisector of the angle. Ex. 14. Find the mass-center of the first quadrant of the ellipse 6^x^ Ex. 16. Find the mass-center of the area between the parabola y'^ = i: ax, and : (a) the double ordinate for x = h; (6) the ordinate for x = li and x-axis. Ex. 16. Show that the mass-center of the circular spandril formed by a quadraat of a circle of radius a and the tangents at its extremities is at a distance . 2234 a from either tangent. Ex. 17. Find the mass-center of a quadrant of the hypocycloid x^ + y^ 2 Ex. 18. Find the mass-center of the area between the parabola x^ + y^ = a* and the axes. x' Ex. 19. Find the mass-center of the area between the cissoid y^ and its asymptote. « — ^ Ex. 20. Find the mass-center of the cardioid r = 2a(l — cos e). Ex. 21. Find the center of mass of the solid paraboloid generated by the revolution of 2/^ = 4 ax about the x-axis. Ex. 22. Show that the center of mass of a solid hemisphere of uniform density and radius a, is at a distance fa from the plane of the base. Ex. 23. Show that the center of mass of a solid hemisphere, radius a, in which the density varies as the distance from the diametral plane is at a dis- tance rf^ a from this plane. Also show that the mean density of this hemi- sphere is equal to the density at a distance f a from the base. Ex. 24. Find the center of mass of a solid hemisphere, radius a, in which the density varies as the distance from the center of the sphere. Ex. 25. Find the center of mass of the solid generated by the revolution of the cardioid c = 2 a (1 — cos ff) about its axis. 80. Moment of inertia. Radius of gyration. If in any system of particles the mass of each particle be multiplied by the square of its distance from a given line, the sum of the products thus obtained is called the moment of inertia of the system about that line. Thus, if wi], wij, •••, be the masses of the several particles, 174 INTEGRAL CALCULUS [Ch. X. ''i; '» •■•) their distances from the line, and / denote the moment of inertia, that is, /= Smr^- (1) In any case in which matter is continuously distributed, as in a solid cylinder, a shell, etc., the matter may be supposed to be divided into small portions, A»ii, A/h,, ••■. By reasoning similar to that employed in the last article, it can be shown that ' = \ r^dm. If matter be supposed to- be distributed uniformly along a line or curve, or upon a curvilinear surface or a plane area, the term " moment of inertia " can also be used in reference to curves, surfaces, and plane areas. Let ]i[ denote the total mass of a body, namely | dm, and I its moment of inertia about a given line or axis. If k satisfies the equation Mk' = /; J- ( r^dm that is, if ft;2 = i = I , ^ {dm f k is called the radius of gyration of the body about the given axis. Ex. 1. Find the moment of inertia of a rectangle of uniform density, "whose sides have the lengths h, d about a line which passes through the center of the rectangle and is parallel to the sides of length b. The density per unit of area will be represented by unity. Let the axes of X and y be taken parallel to the sides of the rectangle, the origin being at the center, and let AB = b, BC = d. 80.] Then APPLICATIONS TO MECHANICS '= ^y'dA d b = j J y^dydx 175 6# 12' This moment of inertia is impor- tant in calculations on beams. Since the mass of the rectangle = bd, ' M 12" k- = ^ = <^ A r <■ -b-- > 1 iX 1 ' i X £j y Fig. 61. Ex. 2. Find the moment of inertia of a very thin circular plate of uniform density of radius a about an axis through its center and perpendicular to its plane. Taking the density as unity per unit of area, / = (r^dm= (r^dA ("C'r^-rdrde Jo Jo Also, M Tra' 2 Ex. 3. Find the moment of inertia about its axis of a right circular cone of height h and base of radius 6, the density being uniform, and m being the mass per unit of volume. The moment of inertia is equal to the sum of the moments of inertia of very thin transverse plates like MS. If OC = x, then, by similar triangles, h 176 INTEGRAL CALCULUS [Ch. X. Hence, if dl denote the moment of inertia of the plate B8 of thickness dx, by Ex. 2, 2 A* Therefore, for the whole cone, 2ft* ■f(^--)* Also, 10 Ex. 4. Find the radius of gyration of a uniform circular wire about its diameter. Ex. 5. Find the moment of inertia of the triangle formed by the axes and a line whose intercepts are a and b, about an axis which passes through the origin, and is at right angles to the plane of the triangle. Ex. 6. Find the radius of gyration about its line of symmetry of an isosceles triangle of base 2 a and altitude h. Ex. 7. Find the moment of inertia about the x-axis of the area between the line and the parabola which both puss through the origin and the point (a, 6), the axis of the parabola being along the x-axis. Ex. 8. Find the moments of inertia of the ellipse b'-x^ + a^y^ = dnfi : (a) about the x-axis ; (6) about the !/-axis ; (c) about an axis that passes through the center of the ellipse and is perpendicular to the plane of the ellipse. Apply the results to the circle x^ + y^ = a^. Ex. 9. Find the moment of inertia of the thin plate in Ex. 1, Art. 79, about the x-axis. Ex. 10. Find the moment of inertia of a homogeneous ellipsoid about the x-axis. Ex. 11. Find the moment of inertia of the surface of a sphere of radius a about a diameter, m being the mass per vmit of surface. Ex. 12. Find the moment of inertia of a solid homogeneous sphere of radius a about a diameter, m being the mass per unit of volume. . Ex. 13. Find the moment of inertia of the semicircular plate described in Ex. 12, Art. 79, about the diameter. Ex. 14. Find the moment of inertia, and the radius of gyration about its axis, of a homogeneous right circular cylinder of length I and radius H, m being the mass per unit of volume. Also about a diameter of one end. CHAPTER XI APPROXIMATE INTEGRATION. INTEGRATION BY MEANS OF SERIES. INTEGRATION BY MEANS OF THE MEASUREMENT OF AREAS 81. Approximate integration. It was remarked in Arts. 4, 8 that in most cases in which a differential f{x) dx is given it is not possible to find the anti-differential. In some of these cases, however, an expression can be found that will approxi- mately represent the indefinite integral | f(x) dx. Even if this cannot be done, it is often possible to determine a value that will very nearly be that of the definite integral j /(,«) dx. Art. 82 explains a method, that of integration in series, by means of which an indefinite integral may be expressed as a function of x in the form of a series that contains an infinite number of terms. An important application of this method to another problem is given in Art. 83. Arts. 84-87 set forth a method, that of measurement of areas, which reduces the evaluation of a definite integral to a mere matter of careful computation. In this connection several formulae for the ap- proximate determination of areas are necessarily considered. 82. Integration in series. When the indefinite integral of a given function, f(x) dx, cannot be found by any of the means thus far considered, one of the most usual and most fruitful methods employed is the following : The function f(x) is de- veloped in a series in ascending or descending powers of x. If this series is convergent within certain limits for x, the series obtained by integrating it term by term is also convergent 177 178 INTEGRAL CALCULUS [Ch. XI, within the same limits.* The greater the number of terms taken the more nearly will the new series represent if{x)dx. Ex. ■• T^-"-' ' '^^ s. 1. Find ("- (1 + x^y By the binomial tlieoreni, (1 + x^)*" " 1 3+1.2 3^ 1.2.3 .3^+ • The second member is convergent for values of x between + 1 and — 1. Integration of both members of (1) gives (2) J dx ^^,j._2 _x^,2_^ x" 2.5.8 x'" " ' l'3.6 1.2 '$2. 11 1.2. 3 '33. 16 (1 + x^) The second member represents the required integral for values of x between + 1 and — 1. It follows from (2) that T: (U _j 2 2-5 1 2.5-8 1 (l+x5)*" ^-^ 1-2 '32. 11 1.2. 3 '33. 16 Ex. 8. Find ie'''dx. Since e' = \+z + — 1- - 1-2 1.2.3 (1) e"^' = 1 + x2 + -^ + - 1-2 1.2.3 which is convergent for all finite values of x. * Suppose that /(x) = ao + aix + «.j.t2 + ... + a„-iX"-'-+ rt„x" + .••. (1) Then r/(x)c;x = a,iX + gig-%«g^+...+""-''^" + ""'^""" + .... f2) -^ ^ ^ 2 3 n n + 1 ^ ' The series in (1) is convergent when ""- is less than unity for all values of u beyond some finite number. The series in (2) is convergent when ^ (1 T ft ^ -'^—i and therefore when " , is less than unity for all values of « II + 1 a„-i a„„i beyond a certain number. Since the convergency of both series depends upon the same condition, the second series is convergent when the firet is convergent. 82-83.] APPROXIMATE INTMGRATION 179 Integration of both members of (1) gives ^^ J 3 1 .2. 6 1 .2 .3.7 1 -2 .3.4.9^ The second member represents the required integral for values of x between + 1 and — 1. It follows from (2) that J-i V 3 1 .2 .5 1 .2-3.7 1 .2 .3.4.9 V Fx 4 r_^_ ^^- *• f— ^— (Putsina;=«.) Ex. 9. f!!^. Ex.8, f aVl - KMa;. "^ * Ex. 10. I i- J/ -^ a; xWl -Ti'dx. Ex. 10. I _ dx. J X Ex.11. pJ^c^x. (Compare Ex. 28, page 98.) J x 83. Expansion of functions by means of integration in series. A function can be developed in series by means of the method described in the last article if the expansion of its derivative is known. The series which represents the function is obtained by integrating the series which represents the derivative, and deter- mining the value of the constant of integration. Ex. 1. Expand tan-i k in a series of ascending powers of x. Diilerentiation and division give (Z. tan-la; = ~^^ = (1 - x^ + x^ - a;6 + ...+(- l)"a;'^» )dx, 1 + x^ which is convergent when x lies between — 1 and + 1. Integi'ating, X2n+1 tan-ix = c + x-i- + — -^+ ... + (-1)"- 3 5 7 2re+l The substitution of for x gives mir = c, m being an integer ; and hence, 0.3 x^ x^ tan-'x = m7r + a — — + — — — + •••. INTEGRAL CALC. — : 13 180 INTEGRAL CALCULUS [Ch. XL This series* can be employed for values of x between — 1 and + 1. It can be used for computing the value of tt. For, on putting x = — ^ therein, it is found that ^^ V^ 6 VS\ 9 45 189 ;' whence, ir = 2V3 fl - 1 + i ^+-"V 1^ 9 45 189 I Ex. 2. Expand f sin"' a; in a series in x ; and compute the value of tt by- putting X = I. Ex.3. Derive t ( e-^'dx = 1 -— + ^^ — +■••, which is *J 1.3 1.2.5 1.2.3.7 convergent for all finite values of x. Ex. 4. Show that log (o + K) = log a + ^ - ^, + ^ - -^ + ■ •• when | a; |< 1 ; and that log (a + a;) = loga + 2 _ -2_ + _^! ^ + ... when |a;|>l. a: 2x2 3x» 4x^ The symbol | x | denotes the absolute value of x. Ex. 5. Derive series for log (1 + x), log (1 — x), log 2, log9. Ex. 6. Develop log (x + Vl + x^) in a series by integrating (1 + x^)~- dx. Ex. 7. § Show that J" Vl-/f^sin-2 2L ^* ^ I2.4 ; ^,2.4.6 )^ / 1.3-(2n-l) ^„y 1 V 2.4...2ra / J' k^ being less than unity. (See Ex. 9, Art. 40.) * It is usually called Gregory's series, after its discoverer, James . Gregory (1638-1675). It was found also by Leibniz (1646-1716). t This expansion is due to Newton (1642-1727), and, by means of it, he computed the value of tt. t This integral is often met in the theory of probabilities, and in certain questions in physics. For the evaluation of I C^ dx when x is greater than unity, see Laurent, Cours d'' Analyse, t. III., § IV., p. 284. For the deriva- tion of \ e-'^dx = | -v^, see Williamson, Integral Calculus, Ex. 4, Art. 116. § This integral is called the "elliptic integral of the first kind." It re- ceived the name elliptic integral from its similarity to the integral in Ex. 8, which represents the length of a quadrantal arc of an ellipse, and is known as "the elliptic integral of the second kind." The integral of the first and second kind are usually denoted by F(k, ^), E(k, (x), substituting b and a for x therein, and calculating ^ (6) — <^ (a). But when it is not possible to find the anti-differential of f(x) dx, recourse must be had to other methods. While, on the one hand, as already shown, areas may be determined by evaluating definite integrals, on the other hand, definite integrals may be evaluated by measuring areas. If the anti-differential of f(x)dx is unknown, the value of I f(x)dx can be found in the following way. Plot the curve y —f(x) from X = a to x=zb, erect the ordinates for which x — a, x= b, and measure the area bounded by the curve, the axis of x, and these ordinates. There are several rules or formnlse for determining areas of this kind. The degree of approximation to absolute correctness depends in general only on the patience of the calculator. These formulae, some of which are usually given in manuals for engineers, are called "formulae for the approximate 182 Integral calculus [Ch. xl. determination of areas," or "formulae for approximate quadra- ture." They may be given the more general title, "forniulm for approximate integration." The two rules most frequently employed, namely, the trapezoidal rule and Simpson's one-third rule, are discussed in Arts. 85, 86, and a rule deduced from them is given in Art. 87. Other rules are given in the Appendix.* It should be observed that only a numerical result is obtained by means of these rules. The knowledge of the value of the definite integral | f(x) dx thus calculated does not give any clue whatever to the expression of the indefinite integral | f(x)dx as a function of x. If the indefinite integral {/(xjdx has been found in the form of a series which is convergent for values of x bet\'\-een a and b, the value of the definite integral j f(x)dx, can be found as accurately as one pleases by taking a suffi- ciently large number of terms. Illustrations of this remark have been given in Exs. 1, 2, Art. 81!, and in Exs. 1, 2, 7, 8, Art. 83. 85. The trapezoidal rule. Let AK be a portion of a curve whose equation may or may not be known; and let LA, TK, be drawn at right angles to the line Y ^---T^ OX It is required to find the area AKTL contained between the curve AK, the line LT, and the perpendicu- lars LA, KT. Divide LT into n parts, each equal to h, and at the points of division erect the perpendiculars MB, NC\ •■-, SH. Draw the chords AB, BC, ■■■, HK. A rule for finding the area of LAKT will now be * See Note E. 84-85.] APPROXIMATE INTEGRATION 183 derived by substituting tbe sum of tlie trapezoidal areas, AM, BN, •■■, HT, for the curvilinear area LAKT; that is, by substituting the boundary made up of the chords AB, BC, ■•■, UK, for the curved line AK. On adding the trapezoidal areas beginning at the left there is obtained, area =|(.4L + BM)+^(BM+ ON) + ■■• +~(HS + KT) = \{AL + 2BM+2CN^ V'^HS + KT) Li = ft(|+l + l + . .. + 1 + 1+1), on writing merely the coefficients of the successive ordinates. This mode of writing will be used also in the rules which follow. The greater the number of parts into which LT is divided, the nearer will the total area of the trapezoids be to the area required. If the equation of the curve is y=f(x), the axes being as in the figure, and OL = a, OT=b, the lengths of the successive ordinates beginning with LA are /(«), /(a + /t), /(a + 2 A), ■••, f{b — K), /(&). If iT is divided into n equal parts, h = , and hence, approximately, £/(a;)dx = ^{/(«)+2/(« + ^) + 2/(a + ^^^^^) + - + 2/(6 - h) + f(b) I . /•lO Ex. 1. Evaluate I ac^dz by this method, taking unit intervals. By the given condition, ft = I ; and hence, n = 10. The successive ordi- nates, since/(a;)=x2, are 0, 1, 4, 9, 10, 25, 36, 49, (34, 81, 100. Hence, approximately, I x'^dx = ^{0 + 100 + 2(1 + 4 + 9 + Hi + 25 + 36 + 49 + 64 + 81)}; = 335. The true value of the integral is — , that is, 333|-. Had the interval to 10 been divided into more than 10 equal parts, the approximation to the true value would have been closer. 184 INTEGRAL CALCULUS [Ch. XI. Ex. 2. Show that the approximate value obtained for the above integral, by making 20 equal intervals, is 333|. Ex. 3. Show that the approximate value of t logioxdx, unit intervals being taken, is 7.yU0231. 86. The parabolic or Simpson's * one-third rule. The parabolic rule for approximating to the value of the area LAKT is derived by substituting parabolic arcs through ABC, CDE, ■■■, GHK, for the arcs of the given curve passing through these points, the axes of the parabolas being vertical, and then summing the areas of the parabolic sections, LABQN, NCDEP, ••-, ROHKT, Fig. 5t. which are thus formed. A parabolic arc, as GDE, will more nearly coincide with the given curve through CDE, than Avill the chords CD, BE. For the purposes of this rule, n the num- ber of equal parts into which LT \s divided must be even, since a parabolic strip is sitbstituted for each of the consecutive pairs of trapezoidal strips ; for example, NCDEP for ND + DP. The area of one of the parabolic strips, say NCDEP will first be found. Through D draw C'E' parallel to the chord CE, and produce NC PE to meet C'E' in C", E'. » Thomas Simpson (1710-1761). 85-86.] APPROXIMATE INTEGRATION 185 The parabolic strip NCDEP= trapezoid NCEP + parabolic segment CDE. The parabolic segment CDE = two thirds of its circumscribing parallelogram CC'E'E. Hence, the parabolic strip NCDEP = NPl^iNG + PE) + l\QD - ^{NC + PE)\-\ = 2h(^NG + iQD + iPE) = ^(NC + 4QD + PE). Application of the latter formula to each of the parabolic strips in order beginning with the first on the left, and addition, gives, approximately, area LAKT = ^(l + 4 + 2 + 4 + 2 + - + 2 + 4 + 1), o in which merely the coefficients of the successive perpendiculars LA, MB, ■■■, TK are written. As in the case of the trapezoidal rule, the greater the number of equal parts into which LT is divided, the more nearly equal will the area thus calciilated be to the true area. If the equation of the curve AK is y =/(»), and OL — a, 0T= b, and LT is divided into n parts, each equal to ~ ' , the n lengths of the successive ordinates, LA, MB, ■■■ TK, are f(a), ffa + ^^^\ ffa + 2 ^^^-\ ■■■, f{b). Hence, on calling these successive lengths, y^, .%, 2/2, ••■ y„, rf(x)dx = ^^(y, + 4.y, + 2y, + iys + 2y^+... J a 6 n + 2 2/„_, + 4 2/„_i + 2/„). 186 INTEGBAL CALCULUS [Ch. XI. Eor the sake of computation, this may be put in the form, r fix) da> = l ^^^ \l (j/o + 2/«) + 2 (Vi + »/3 + - + Vn-i) + (V2 + yi+ - + J/n-2)J- Ex. 1. Evaluate \ x* At by this method, taking n = 10. Here, yo, yi, ;/.j, ■•■, yio, are 0, 1, 81, 625, •■•, 10,000, respectively; and hence, approximately, r'x«rfx=|§{iif^+2(l + 81+625+2401 + 6561) + (16 + 256 + 1296 + 4096)} = 20001|. The true value of the given integral is 20,000 ; thus the error is only 1| in 20,000. /■12 Ex. 2. Show that the value o£ ( logioxdx calculated by this rule for n = 10, is 8.004704 (compare Ex. .'3, Art. 85). A comparison betvreen these two rules is given in the following quotation : f " The increase in accuracy (of the parabolic) over the trapezoidal rule is usually quite notable, unless the number of ordinates become large, in which case they both approximate more and more closely to the true value and to each other. In a * If re he the number of equal intervals into which the range 6 — a is divided, the outside limit of error that the parabolic formula for integration can have, is r 6-a \ V'(Xr) . n.i /6-a\ V"(Xr: \^ 2 ) 90 re* in which x, is some value of x between a and 6, and /'" (x) denotes the fourth derivative of /(x). The outside limit of error in the case of the trapezoidal rule is 12 n2 -^ ^ '■" in which f"{x) denotes the second derivative of /(x). If re is doubled, the limit of error is reduced, therefore, to ^ and \ of its former amount. (See Boussinesq, Coiirs W Analyse, t. II. 1, § 262, and Markoff, Differenzen- rechnung, § 14, pp. 57, 59.) t This is from an article, entitled, " New Rules for Approximate Integra- tion," in the Engineering News (N. Y.), January 18, 1894, by Professor W. E. Durand of Cornell University. 86-87.] APPROXIMATE INTEGRATION 187 series of trials made by the author upon a number of integrals of various forms for the purpose of testing the relative accuracy of these rules, it was found for cases in which the locus was of single curvature only that the trapezoidal rule required about double the number of sections for equal accuracy with the parabolic rule. \Vhere the locus involves several changes of curvature, as in lumpy and irregular curves, and the number of sections is moder- ate, one rule is as likely to be right as the other, and both are likely to be considerably in error. For a large number of sec- tions, however, the parabolic rule will show its superiority as above." 87. Durand's rule. From a discussion * on the trapezoidal and parabolic rules, Professor Durand has deduced another rule for which " it seems not unfair to claim substantially the full prob- able accuracy of the parabolic rule, and practically the simplicity in use of the trapezoidal rule." It is as follows, merely the co- efficients of the successive ordinates being written in order from the left : approximate area = h [^ -|-|| + 1 + 1+ ••• -f-l-fl-ff|-f- j^] ; or, approximately, area = h[A + 1.1 + 1 + 1 +•" +1 + 1 + 1.1 + .4]. The number of intervals may be even or odd. J" 60° sin e de with 10° intervals. The circular measure of 10° is .17453. The rule gives for the approximate value of the integral, r*'°sin Bde = .1745.3 [.4 (sin 0° -|- sin 60°) -f 1.1 (sin 10° + sin 50°) " -I- (sin 20° -f sin 30° -|- sin 40°) ] = .5000075. Since the exact value of i sin 9 d9 is - cos 9 , or .5, the difference between the above approximate and the true values of this integral is not more than one part in 66,666. * In the article mentioned in the preceding footnote. 188 IXIEGRAL CALCULUS [Ch. XI. rw Ex. 2. Show that I x- dx calculated by this rule with unit intervals gives -'0 a difference of one jiart in o.>33. Ex. 3. Show that I logio x calculated by this rule with unit intervals is 8.004062. (Compare with Ex. 3, Art. 85, and Ex. 2, Art. 86.) 88. The planimeter. Attention has been drawn to the fact that the value of a definite integral is also the value of a certain plane area, and that, consequent!}', the measurement of the area is equivalent to the evaluation of the integral. In Arts. 85, 86, S", rules are given for approximately determining plane areas, and other rules therefor are given in the Appendix.* These areas can be measured exactly by instruments called mechanical integrators or planimeters. A planimeter measures the area of any plane figure by the passage of a tracer round about the perimeter of the figure, the readings being given by a self-recording apparatus. There are several kinds of planimeters, but they all have certain fundamental properties in common. The first planimeter was invented by the Bavarian engineer, J. M. Hermann, in 1814. Amsler's polar planimeter, which was invented by Jacob Amsler when a student at Konigsberg in 1854, is the most popular on account of its simplicity and handiness in use. Thousands of them have been made at his works in Schaffhausen. The Amsler planimeter is shown in Fig. oo. It consists of two bars, (a) the radius bar, and (6) the pole arm, jointed at the point C. The tracing point P, which now coincides with the point B of the figure ABDE, is carried round the curve, and the roller vi, which partljr rolls and partly slips, gives the area of the figure ; and by means of the graduated dial h, and the vernier v in con- nection with the roller m, the result is given correctly in four figures. The sleeve H can be placed in different positions along the pole-arm b, and fixed by a screw s so as to give readings in different required units. A weight at iv is placed upon the bar to * See Note E. 87-88.] APPROXIMATE INTEGRATION 189 keep the needle point in its place, but in instruments by some other makers T is a pivot in a much larger weight, which rests on the paper. The accuracy of the reading depends upon the accuracy with which the tracing point follows the curve.* Fig. 55. Professor O. Henrici's Report on Ptanimeters (Repor.t oi the British Association for the Advancement of Science, 1894, pp. 496-523) contains a slfetch of the history of planimeters, the geometrical theory of gener- ating areas, descriptions of early planimeters, a discussion on Amsler's planimeter, and a description of some recent planimeters. Professor H. S. Hele Siiaw's paper on Mechanical Integrators (Proceedings of the Institution of Civil Engineers, Vol. 82, 1885, pp. 7-5-143) gives an account of the theory and the practical advantages of several varieties of planimeters. The descrip- tion given above is from this paper. An explanation of the theory of Amsler's planimeter is given by Mr. J. IWacFarlane Gray in Carr's Synopsis of Mathematics. There is a discussion on planimeters in Professor R. C. Carpenter's Text-book of Experimental Engineering, pp. 24-49. * For the fundamental theory of the planimeter, see Note P, Appendix. CHAPTER XII INTEGRAL CURVES 89. Introduction. A first integral curve was defined in Art. 15. The student is advised to review that article thoroughly before proceeding further. In this chapter the subject of integral curves will be studied more fully, and some of their applications to mechanics will be pointed out. Differentiation under the sign of integration is an important topic in the integral calculus. Only a very special case, however, is necessary in what follows : this case is considered in Art. 90. Arts. 92, 93, 94, contain an exposition of the simpler properties of integral curves and a few examples of their usefulness. Their applications are of especial value to the student of engineering. For the joroper understand- ing of several of them, a better acquaintance with the theorems of mechanics is required tlran some readers of the calculus ma}' be presumed to have at this stage. Accordingly, a further expo- sition of the service that may be rendered by these curves is given in the Appendix for purposes of future reference. Articles 94, 95, discuss the practical plotting of integral curves.* 90. Special case of differentiation under the sign of integration. A special case of differentiation under the sign of integration * Arts. 91-95 and the related matter in the Appendix are taken witli some slight but no essential change, from an article entitled Inlerjral Curoes, by Professor W. F. Durand, Principal of the Graduate School of Marine Engineering and Naval Architecture, Cornell University. The article, which appeared in the Sihley Journal of Engineering, January, 1897, is practi- cally all reproduced here. This chapter has also had the benefit of Professor Durand's revision. 190 89-90.] INTEGRAL CtlRVES 191 with, respect to one of the limits which is also involved in the function under the sign will be considered. Let /= CQ)-xyf{x)dx. (1) Jo The differential coefficient — will be derived by the funda- mental method employed in differentiation, namely, by giving an increment to the variable, in this case h, then finding the cor- responding increment in /, and finally obtaining the ratio of these two increments when the increment of b approaches zero. Suppose that & receives an increment A&, then from (1) (6 + A& - £»)»/(») dx. I) Hence J'»6+A5 /»6 (& + A6 - xyf{x) dx - I (& - xyf(x) dx ; Jq whence, by Art. 7 (b), X6 /•6+A& (& + A& - xyf{x) dx+ t ■ {b + Ab- «)"/(«) '''■«• - r(b-xff{x)dx= r [(6 + A6-a;)»- (6 -£»)"]/(»;) da- »/o Jo X6+A& (b + Ab- xYf{x) dx. From this, by Art. 7 (c), A/= C lib + A6 - a;)" - (b — a;)"]/(a;) dx Jo + Ab\b + Ab ~ {b + e ■ Ab)\ "/(b + $ ■ Ab), in which 6 <1. Hence, remarking that A6 is independent of x, and can there- fore be put under the integration sign, ^I^r[M^tz^rz:^^^nf(ai)dx+\{l-e)Ah\"f{b + d.Ab). Ab Jo Ab 192 INTEGRAL CALCULUS [Ch. XII. Therefore, letting A6 approach zero, f^ = nj\b-xr-^f{ai)dx. (2) This result will be required in Art. 93. 91. Integral curves defined. Their analytical relations. A more general definition of an integral curve than that given in Art. 15 will now be introduced. In what follows, a number of curves will be spoken of together. In order to distinguish between them, the system of ordinates, that is, the y's, for each of the several curves will be denoted by a subscript number. If y=m, or, for the sake of distinction, 2/0 =/(«:) (1) be the equation of a given curve, the curve whose equation is . 2/i = - ( yo<^^ (2) is called & first integral curve of the curve whose equation is (1). The latter is called the fundamental curve. Since | y dx is of the second dimension, and y^ should be linear, the constant factor - is introduced in (2), in which a is a linear quantity and has a magnitude that will make equation (2) convenient for plotting. It may be called a sccde factor. In the definition in Art. 15 the scale factor was unity. From (2) on differentiation, -J- = - 2/0- dx a Hence, as x varies, the slopes of the first integral curve vary as the ordinates of the fundamental ; and therefore the former can be represented by the latter, and vice versa. 90-91.] INTEGRAL CURVES 19p The first integral curve (2) also has a first integral, the latter has a first integral, and so on. These successive integral curves are called the second, third, etc., integral curves 'of the original or fundamental curve. On using the constant linear quantities, b, ■ c, ■■■ w, as scale factors for the sake of plotting the curves con- veniently, and on distinguishing by different subscripts the ordi- nates that belong to the various curves, the latter will have the following equations : Fundamental, 2/0 =/(«); (1) 1 C^ first integral, 2/1 = - I Vod^i (2) a Jo second integral, 2/2 = 7 I yidx = ~- \ I yodx^; (3) bJo abJii Jo third integral, y-i=- \ yidx = —- \ ( ( y^dx^; (4) cJa abcJo Ja Jo nth. integral curve, 2/„ = - Cyn-, dx = — i— CrC-Cyo dx\ (5) tvJo abc---wJo Jo Jo Jo From equation (2) dVn 1 '^ C" 1 d^y.2 1 /n\ and hence, ;p;^ = 7^y<'- *■ '' And in general, da^ ab' d"y„ clx" abc ■■•IV 2/0. (8) Equation (8) shows that as x varies, the wth derivatives of the nth integral curve vary as the ordinates of the fundamental curve ; and therefore, the former can be represented by the latter. 194 INTEGRAL CALCULUS [Ch. XII. 92. Simple geometrical relations of integral curves. In Fig. 56 BP, OA, OB, OC, represent the fundamental, and the first, second, and third integral curves respectively, whose equations are (1), (2), (3), (4), of Art. 91. ,.;4sr2^ (a) As X increases, and so long as the fundamental curve BP lies above the «-axis, the ordinates of the first integral OA will increase, and the tangent to OA will make a positive angle with the a^axis ; when BP lies below the a^axis the tangent to OA makes a negative angle with the x-axis ; when BP crosses the a^axis, the tangent to OA is parallel to the a;-axis. These prop- erties follow from equations (2) and (6), Art. 91. (b) At points for which the ordinate of the first integral curve is a maximum or a minimum, and there also, by (6), #1 dx 2/0 = 0; :0. Hence, to a zero value of the ordinate of the fundamental there corresponds a maximum or a minimum value of an ordinate of the first integral curve. 92-93.] INTEGRAL CURVES 195 (c) At points where an ordinate of the fundamental is a maxi- mum or a minimum, dx ' and at points where the first integral curve has a point of in- flexion, Differentiation of the members of equation (6) shows that ^=Owhen^l!'=0. ax- ax Hence, to a point on the fundamental at which there is a maxinuim or a minimum value of the ordinate, there corresponds a point of inflexion on the first integral curve. 93. Simple mechanical relations and applications of integral curves. Successive moments of an area about a line. If each in- finitesimal portion of a plane area be multiislied by its distance from a given line, the sum of a;ll these products is called the mome)it of first degree of the area about the line. If each of the infinitesimal portions of the area be multiplied by the square of its distance from the given line, the sum of all the products is called the moment of second degree of the area with respect to the line. The latter is the moment of inertia of the area about the line, examples of which were shown in Art. 80. The moment of first degree is usually called the statical moment. In general, if each infinitesimal portion of an area be multiplied by the ?ith power of its distance from a given line, the sum of all these products is called the moment of the nth degree of the area about the line. For the sake of brevity, this may be called the nth moment. Thus (Fig. 56), lay off OX = x^, and erect the ordinate AP at X, and consider | ^y^dx, the area OBPX, between the funda- INTEGRAL CALC. — 14 196 INTEGRAL CALCULUS [Ch. XII. mental, the axes, and the ordinate AP. If the successive moments of this area be taken about the ordinate AP for which x = x^, and these moments be denoted by Mi, M^, ■■•, M^, in order, then the first moment. Mi— \ \xi — x^y dx; (1) •Jo (xi — xfy dx ; (2) the nth moment, M„= I ^{xi — xyydx. (3) (a,'] — xyy dx, = j V dx, the area. (4) Jo (a) Differentiation of (3) with respect to Xi will give by equa- tion (2), Art. 90, rfJ/„ f'v Nn-l ^ = n \ (Xi — xY ^y dx ; dxi Jo that is, ^ = nM„^i. (5) dxi Hence, M,, = n{ 'M„ _i dxi. Jo Since dxi is an infinitesimal distance along the a;-axis, it can be written dx, and hence M„ = nC ^M„_idx. (6) By successive application of (6) there will finally be obtained, J/„ = w! pp.. pModx". (7) That is, the nth moment of the area ORPX about an ordinate distant x^ from the origin is equal to factorial n times the »th 93.] INTEGRAL CURVES 197 integral of the moment of degree zero for the same area. On substituting for M^ in (7) its value from (4), there is obtained, ) J" J Jo y^'^'^y*^- (^) Hence, the value of the nth moment of the area of y =/(«) above described about an ordinate distant x^ from the origin, is equal to factorial n times the ordinate of the (n + l)th integral curve at a; = a-j ; and, therefore, the ?ith moment may be repre- sented by this ordinate. On using F„.,.i to denote the ordinate of • the (n + l)th integral curve at a; = x^, this may be expressed by In particular, the statical moment (1) is represented by the corresponding ordinate of the second integral curve, and the mo- ment, of inertia (2) by twice the corresponding ordinate of the third integral curve. Thus in Fig. 56, the area ORPX is represented by AX; its statical moment about ^Pis represented by BX; and its moment of inertia about AP by 2 GX. Suppose that the scale factors used in plotting the threfe inte- gral curves, each from the one of next lower order, are a, b, c, respectively, as indicated in equations (2), (3), (4), Art. 91. Then, a.vea.ORPX=a- AX; the statical moment of ORPX about AP = ah ■ BX; the moment of inertia of ORPX about AP = 2 ahc ■ OX. (b) If G is the center of mass (or center of gravity) of ORPX, its distance HX from AP, by Art. 79, is determined thus : £\^^-^)y'i^ ^M,_aby,^,BX_ c, M„ ay^ AX ydx £ 198 lyTEGRAL CALCULUS [Ch. XII. (c) If k is the radius of gyration of the area OBPX about AP, then by (Art. 80), ,,2 _ Moment of Inertia about AP _ 2 abojs __ r, , O^ . Area ay, AX For further applications to mechanics, and some general re- marks on the use of these curves in engineering problems, see Appendix. The reader is recommended to glance at the latter remarks now. 94. Practical determination of an integral curve from its funda- mental curve. The integraph. Suppose that the equation of the fundamental curve is 'j = f{x). The ordinates of the first integral curve that correspond to successive values .i\, x^, ■•■, x„, of the abscissas are 1 r'i 1 z*"^ 1 /"^ - yd^, - ydx, ■■;- ydx, nJo a .Jo « »/o respectively. These may be determined by the ordinary rules for integration when the functions rnider the sign of integration are integrable. If the latter condition does not hold, recourse can be had to some of the various methods of mechanical and approximate integration described in Arts. So-88. It will be necessary to do this also, ■when the fundamental has been plotted merely from a knowledge of the ordinates that correspond to particidar abscissas, the equation of the curve being unknown. For example, in Fig. 67, the area of each successive section between the ordinates of the fundamental may be found with a planimeter, and the ordinates of the integral curve, which is shown by the dotted line, may be found by successive additions. As an instrumental check, it is well from time to time to go around the entire area between the 2/-axis and the ordinate in question, and compare the residt with the total area summed to that point. Numerical means of integration may also be em- ployed. The trapezoidal rule and the parabolic rule can be readily used for finding successive increments of area in the case 93-94.] INTEGRAL CURVES 199 of the fundamental, and hence for finding successive increments of the ordinates of the first integral curve. In whatever way the integral curve may be derived from the fundamental, it is well, after plotting, to compare the two and note the fulfillment of the simple geometrical relations, (a), (b). Y i -- . ,___ 3 ^y "-. .' ■■- ^ ■" ^ / ''■^ ^-'. 2 / / \ ^'^ --. _ .^- ^ / f \ / 1 / \ ^ ' ^ / / / \ s ,■' / \ X \ / \ 1 \ / \ \ / \ ^ -^ " 3 4 (c), of Art. 92. Thus, one should look for a maximum or a mini- mum ordinate in the integral corresponding to every zero ordinate in the fundamental, and for a point of inflexion in the integral for each maximum or minimum in the fundamental. The tan- gent of the integral varies with the ordinate of the fundamental, and hence, the slope of the integral should increase or decrease when the ordinate of the fundamental increases or decreases. These relations may be noted in the curves in Figs. 56, 57. The intpgrapli is an instrument that is used for drawing the first integral curve from its fundamental. The theory of it is given in the Appendix (Note G). It ■ may be used also for determining. the area between a curve and the a;-axis. For the 200 INTEGRAL CALCULUS [Ch. XII. integral curve can be drawn with the integraph, and the ordinate corresponding to the area can be measured. Since the length of the ordinate represents the area, the latter can be found immediately on making allowance for the scale-factor. 95. The determination of scales. In order to have the various curves convenient for plotting, it is usually necessary to employ different scales for the ordinates. If numerical integration is used, the value of the area of the fundamental will be found directly, and the scale may be correspondingly selected so that the curve will be kept within the desired limits as to size. If the planimeter is used, the result will be given in square inches or other area units, and must be converted into the value desired by the use of a scale factor. Suppose the fundamental plotted as^ follows : horizontally 1 unit of length = p units of abscissa, vertically 1 unit of length = q units of ordinates. Then 1 unit of area on the diagram will represent pq units of the integrated function, and the area found must be multiplied by this factor in order to reduce it to the value of the integral desired. The scale factors a, b, c, etc., may then be chosen as before. * See Note G. CHAPTER XIH ORDINARY DIFFERENTIAL EQUATIONS 96. Differential equation, order, degree. A few differential equations which frequently appear in practical work will now be discussed very briefly.* A differential equation is an equation that involves differentials or differential coefficients. Ordinary differential equations are those which contain only one independent variable. For example, dy = cos X dx, (1) dl = <^' (2) (3) (4) (5) are ordinary differential equations. The order of a differential equation is the order of the highest derivative that appears in it. The degree of a differential equa- tion is the degree of the highest derivative when the equation is * For fuller explanations than are given here, reference may be made to Introductory Course in Differential Equations, by D. A. Murray. (Long- mans, Green, & Co.) 201 B-. {- +(I)T cPy da? y = dx dx (2/+cy idx dz^ dy dz .(y + a)=0, 202 INTEGRAL CALCULUS [Ch. XIII. free from radicals and fractions. Of the examples above, (1) is of the first order and first degree, (2) is of the second order and first degree, (3) is of the second order and second degree, (4) is of the first order and second degree, (5) is of the first order an.l first degree. Differential equations of a very simple kind have already been considered. 97. Constants of integration. General and particular solutions. Derivation of a differential equation. If a relation between the variables together with the derivatives obtained therefrom satis- fies a differential equation, the relation is called a solution or integral of the differential equation. For example, y = m sin x, (1) y = n cos X, (2 ) y = A cos X -\- B sin x, (3 , y = c sin (x + a), (4 in which m, )i, A, B, c, a, are arbitrary constants, are all solutiou- of the equation This may be verified by substitution. It will be observed that (.")) does not contain m, n, A. B, c, or u.. The solutions of the differential equations of the first order which have appeared in the former part of this book contain one constant of integration ; those of the second order contain two constants. Examples have been given in Arts. 8, 9, 12, 59, etc. Solutions (1) and (2) above contain one arbitrary constant, and solutions (3) and (4) each contain two. The question is sug- gested : Hoiv many arbitrary constants should the most general solution of a differential equation contain ? The answer can in part be inferred from a consideration of one of the ways in which a differential equation may arise, namely, by the elimination of constants. On comparing (3) and (5) it is seen that (5) must 'J6-Q7.T- ORDINARY DIFFEHENTIAL EqUATlONS 203 have been derived from (3) by the eliminatioa of the two con- stants A and B. In order to eliminate two quantities, three equations are necessary. One of these is given, and the others can be obtained by successive differentiation. Thus, y = A cos x-{- B sin x, -^ = — Asva.x -\- Bcosx, dx -—-„ = — A cos a; — i? sin a; : dar whence, — ^ 4- w = 0. da^ ^ In order to eliminate three constants from a given equation, four equations are required. Of these, one is given and the remaining three can be obtained only by successive differentia- tion. The third differentiation will introduce a differential coeffi- cient of the third order, which accordingly will appear in the differential equation that is formed by the elimination of the three constants. In general, if an integral relation contains n arbitrary constants, these constants can be eliminated by means of « -|- 1 equations. The latter consist of the given equation and n rela- tions obtained by n successive differentiations. The nth differ- entiation introduces a differential coefficient of the nth order, which will accordingly appear in the differential equation that arises on the elimination of the constants. The solution of an equation of the nth order cannot contain more than v constants ; for if it had n + 1, their elimination would lead to the equation of the n + 1th order.* The solution that contains a number of arbitrary constants equal to the order of the equation is called the general solution or the complete integral. Solutions obtained therefrom by giving * For a proof that the general solution of a differential equation contains exactly n arbitrary con.stants, see Introductory Course in Differential Equa- tions, Art. 3 and Note C, Appendix. 204 INTEGRAL CALCULUS [Ch. Xlll. particular values to the constants are called parliudar solutions. For example, (3) and (4) are general solutions of (6), and y = 2 cos X + 3 sin x, y = 5 cos x — sin x, y = 2 sin (x + w), y = 3sinfx-^\ are particular solutions. Ex. 1. Eliminate the arbitrary constants m and c from (1) y — mx + c. Differentiating twice, (2) ^ = ni, dz (3)g=0. These equations may be interpreted geometrically. If m, c, are both arbitrary, (1) is the equation of any straight line ; and, therefore, (3) is the differential equation of all straight lines. If c is arbitrary and m has a definite value, (1) is the equation of any line that has the slope to, and, accordingly, (2) is the differential equation of all the straight lines that have the slope m. Ex. 2. Find the differential equation of all circles of radius r. The equation of any circle of radius r is (x-ay+(y-b)^ = r^ in which a, b, the coordinates of the center, are arbitrary. The elimination of a and 6 gives the equation required. Ex. 3. If !/ = Ja;2 + B, prove that x^-^ = 0. dx^ dx Ex. 4. Eliminate c from y = ex + c — c^. Ex. 6. Form the differential equation of which e'^n + 2 cxe>' + c^ = is the complete integral. Ex. 6. Eliminate the constants from y = ax+ bx^. Ex. 7. Form the differential equation which has y = a cos {mx + b) for its complete integral, a and 6 being arbitrary constants. 97-99.] ORDiNARY DlFfERENrlAL EQUATIONS 205 Section I. Equations of the First Oedek and the First Degree. 98. Equations in which the variables are easily separable. If an equation is in the form Ux)dx+f,(y)dy = 0, its solution, obtainable at once by integration, is jfiix) dx +jf2(y) dy = c. Some equations can easily be put in this form. Ex. 1. Solve (1 - »)d2/ -(1 + y^dx = 0. This may be written, -^ ^ = 0. 1 + 2/ l-x This step is called "separation of the variables." Integrating, log {1 + y) + log (1 - x) = a, or, (1 + j/)(l -a;) =e=i = c. Ex.2. Solve ^ + JL:^ = o. ax ^i-x^ Ex. 3. Solve 3e'ta,nydx + (l — e") sec^ ydy = 0. 99. Equations homogeneous in x and y. If an equation is homo- geneous in X and y, the substitution y = vx will give a differential equation in v and x in which the variables are easily separable. Ex. 1. Solve (»2 + y2-)dx -2xydy = 0. Rearranging, (1) ^ = 2i±i^. dx 2xy On putting y = vx, ^ = v + x^. dx dx 206 INTEGRAL CALCULUS [Ch. XIIl. Substitution of these values in (1) gives V + X — = dx 2v Separating the variables, = 0. X 1 —v^ Integrating, log x( 1 — v^) = log c ; that is, loga;n - ^ ] = logc ; whence, x^ — y^ = ex. Ex. 2. Solve j/2 dx + {zy + x^) dy = 0. Ex. 3. Solve oi'^y dx — (x^ + y^) dy = 0. Ex. 4. Show that the non-homogeneous equation of the first degi-ee in x and y dy_ ax + by + c dx a'x + b'y + c' is made homogeneous, and therefore integrable, by the substitution x = Xi + h, !/ = ,Vi + k, h, k being solutions of ah + bk + c = 0, a'h + b'k + c' = 0. 100. Exact differential equations. An exact differential equa- tion is one that is formed by equating an exact differential to zero. It follows from Art. 24 that Mdx + Ndy = is an exact differential equation if dM^dN dy dx Ex. 1. Solve (a' ~2xy- if-) dx - (x + ij^dy = 0. Ex. 2. Solve (x2 - 4 xj/ - 2 «/) dx + (y"^ - ixy -2x'')dy = 0. Ex. 3. Solve (2 x^y + 4 x^ - 12 xi/^ -|- 3 !/2 _ xe' + e-»^) dy + (12 x'-(/ + 2 XJ/2 + 4 x8 - 4 j/3 + 2 ye-' - ef) dx = 0. 99-101.] ORDINARY DIFFERENTIAL EQUATIONS 207 101. Equations made exact by means of integrating factors. The differential equation ydx — xdy = is not exact. Multiplication by — gives y ydx — xdy y' This is exact, and its solution is - = c, or x= cy. y When multiplied by — > the first equation becomes xy dx dy _ X y which is also exact. The solution is log X — log y = log c, whence, - = Cj ot x = cy. y Another factor that will make the given equation exact is — • Any factor such as — , — , — employed above, which changes x' y^ xy x^ an equation into an exact differential equation, is called an integrating factor. It can be shown that the number of integrat- ing factors is infinite. There are several rules for finding integrating factors. In the following examples, the necessary integrating factor can be found by inspection. Ex.1. Sohie ydx — xdy +\o^xdx = 0. Here, logxcfe is integrable, but a factor is needed for ydx — xdy. Obviously — is the factor to be employed, as it will not affect the third term a;2 -. injuriously from the point of view of integration. On multiplication by — x^ the given equation becomes ydx — xdy . loga: _ q 208 INTEGRAL CALCULUS [Ch. XIII. The solution of this equation reduces to ex + y — logic — 1=0. Ex. 2. a{xdy + 2ydx) = xydy. Ex.3. {x^ + y'' + l)dx-2xydy = 0. Ex. 4. (x^e' — 2 my'') dx + 2mxydy = 0. 102. Linear equations. If the dependent variable and its de- rivative appear only in the first degree in ■a, differential equation, the latter is said to be linear. The form of the linear equation of the first order is | + P.= Q, (1) in which P and Q are functions of x, or constants. The linear equation occurs very frequently. The solution of dx ^ ' that is, of — = — Pdx, y - [pdx \pdx IS 2^ = ce ■* , or ye^ = c. On differentiation the latter form gives J''^(dy + Py dx) = 0, which shows that e'"" ' is an integrating factor of (1). Multiplication of (1) by this factor gives e''''"(dy + Py dx) = e^'''^Qdx; and this, on integration, gives or y^e~^'"'''{^J'"'-Qd^ + c}. (2) 101-103.] OBDiyABT DIFFERENTIAL EQUATIONS 209 The latter can be used as a formula for obtaining the value of y in a linear equation of the form (1). Ex. 1. Solve x^-y =xK fix This is linear since y and -^ appear only in the first degree. On putting it in the ordinary form (1), it becomes dx X y^^-.-n Here P = — , and hence, the integrating factor e = e ' = e ^^S = -. x X By using this factor, and adopting the differential form, the equation is changed into -dy ydx = xdx. X- x^ w 1 1 Integrating, ^ = -x'^ + c, or a = -x^ + ex. X 2 2 Ex. 2. Solve ^ + 2/ = e-«. dx Ex. 3. Solve ^ + ^ ~^^ y = 1. dx x^ Ex.4. Solve ^ + -i^-2/ = i -■ dx x^ + 1 (a;2 + l)3 Ex.5. Solve ^ + ^2,=^. dx X x" 103. Equations reducible to the linear form. Sometimes equa- tions that are not linear can be reduced to the linear form. One type of such equations is ^ + Py=Qr, dx in which P, Q, are functions of x, and n is any constant. Divi- sion by 2/" and multiplication by {—ii + 1) gives i-n + l) y-"^ +{-n + l) Py-+^ = (- n + 1) Q. 210 INTEGRAL CALCULUS [Ch. Xlll. On substituting v for y~"^^, this reduces to ax which is linear in v. Ex. 1. Solve ^ + iy = x'^y^. dx X Division by y^ gives y-^ 2^ _)_ _ y-i — x^. dx X On putting v for y^, this takes the linear form V =— ox*^. dx X The solution Is u = y-^ = cx^ -\- ^x^. Ex.2. Solve ^ + -j/ = 3x2yi dx X V.X.3. Solve ^ + ^L = xyi (Ix 1 — x'^ E.X.4. Solve 3^+^-2,=^. dx x+ 1 y' Ex. 5. Show that the equation dx ill which P, Q, are functions of x, becomes linear on the substitution of v for /(!/). Section II. Equations of the First Order but not of THE First Degree. 104. Equations that can be resolved into component equations of the first degree. In what follows, ^ will be denoted by p. The da; type of the equation of the first order and 7ith degree is which on expansion becomes P" + I\p"-' + P,p-~^ + . . . + p„_j p + p^^y = 0. (1) 103-105.] ORDINARY DIFFERENTIAL EQUATIONS 211 Suppose that the first ^ member of (1) can be resolved into rational factors of the first degree so that (1) takes the form, {p-E,)(p-B,)--(p-li„) = 0. (2) Equation. (1) is satisfied by any values of y that will make a factor of the first member of (2) equal to zero. Therefore, in order to obtain the solutions of (1), equate each of the factors in (2) to zero, and find the integrals of the n equations thus formed. Suppose that the solutions derived from (2) are /i i^, y, Ci) = 0, /s (x, 2/, C2) = 0, • • •, /„ («, y, c„) = 0, in which c„ c.,, •••, c„, are arbitrary constants of integration. These solutions are just as general if «, = 03= ■•• = c„, since each of the c's can take any one of an infinite number of values. The solu- tions will then be written M^,y,o)=0, f^ix, y, c) = 0, ■■; f„(x,y,c) = 0, or simply, /i {x, y, g)/^ (x, y,c)---f„ {x, y, c) = 0. Idyy \dxj Ex.1. Solve (!^]" + (x-|-j/)^ + X2/ = 0. -'- dx This equation can be written {p + y){p + x) = (i. The component equations are p -\- y = 0, p + x = 0, o£ which the solutions are log !/ + a; + c = 0, 2 y + x^ + i c = 0. The combined solution is (log y Jr x + c){2 y + x'^ -\- 2 c) = 0. Ex. 2. Solve ( ;p I = a^*- Ex. 3. Solve jfi + i xp^ - 2/V - 2 xy'^p = 0. 105. Equations solvable for y. When equation (1), Art. 104, cannot be resolved into component equations, it may be solvable for y. In this case, f{x, y,p) = can be put in the form y = F(x, p). Differentiation with respect to x gives INTEGRAL CALC. — 15 212 INTEGBAL CALCULUS [Ch. XIII. which is an equation in two variables x and p. From this it inay be possible to deduce a relation The elimination of p between the latter and the original equa- tion gives a relation that involves x, y, c. This is the solution required. If the elimination of p is not easily practicable, the values of x and y in terms of p as a parameter can be found, and these together will constitute the solution. Ex. 1. Solve X— yp = ap^. „ X — ap^ Here y — P Differentiating with respect to x, and clearing of fractions, This can be put in the linear form dx 1 _ ap dp pQ. -p») 1 -i)2 Solving, X = ^ (c + a sin-ip). Vl-^2 Substituting in the value of y above, y = -ap-\ — (,c + a sin-ip), Ex. 2. Solve 42/ = a;2 + p2. Ex. 3. Solve !/ = 2p + 3p2 106. Equations solvable for as. In this ease /(«, y, p) can be put in the form x==F{y,p). Differentiation with respect to y gives from which a relation between p and y may sometimes be obtained, say, f{y,p,c)=0. 105-107.] ORDINARY DIFFERENTIAL EQUATIONS 213 Between this and the given equation p may be eliminated, or x -and y may be expressed in terms of p as in the last article. Ex. 1. Solve x = y + a logp. Ex. 2. Solve x(l +p2)=l. 107. Clairaut's equation. Any differential equation of the first order which is in the first degree in x and y comes under the cases discussed in Arts. 105, 106. An important equation of this kind is that of Clairaut.* It has the form y=px+f(p). (1) Differentiation with respect to x gives From this, p = = P+{x+f{p)l dp dx «>+f'(p)-- = 0, (2) dp_ dx = 0. (3) nation it follows that » = c •: Substitution of this value in the given equation shows that y = cx +f{c) (4) is the general solution. See Introductory course in differential equations, Art. 28, for remark on equation (2). Some equations are reducible to Clairaut's form, for instance, Ex. 2 beloyv. Solution (4) represents a family of straight lines. The en- velope of this family of lines will also satisfy the differential equation, since x, y, p, at any point on the envelope is identical with the X, y, p of some point on one of the tangent lines of which (4) is the equation. The equation of the envelope of (4) is called the singular solution of (1). Singular solutions are discussed in Chapter IV. of the work referred to above. * Alexis Claude Clairaut (1713-1765) was the first person who had the Idea of aiding the integration of differential equations by differentiating them. He applied it to the equation that now bears his name, and published the method in 1734. 214 INTEGRAL CALCULUS [Ch. XIII. Ex. 1. Solve y = px + aVl + p'^. The general solution, obtained by the substitution of c for p, is which represents a family of lines. The envelope of this family is the circk x2 + y2 _ 32. The latter equation is the singular solution. Ex. 2. x'^{y —px) = yp"^. On putting x- = u, y- = v, the equation becomes u = M — + /' — V du \duj vyhich is Clairaut's form. The solution is V = cu + c^, that is, y^ = cx^ + c2. Ex. 3. y =px + sin-^p. Ex. 4. py =p-x + '»• Ex. 5. X(/' = pi/x^ + x +py. 108. Geometrical applications. Orthogonal trajectories. A curve is often defined by some property whose expression takes tlie form of a differential ecjuation. In the examples given below the differential equations of the curves are of a less simple character than those which appeared in similar problems in Arts. 8, 12, 32. Problems that relate to orthogonal trajectories are of consider- able importance. Suppose that there is a singly infinite system of curves f(x, y, a) = 0, (1) in which a is a variable parameter.^ The curves which cut all the curves of the given system at right angles are called orthogonal trajectories of the system. The elimination of a from (1) gives an equation of the form 0(..y,|) = O, (2) 107-108.] ORDINARY DIFFERENTIAL EQUATIONS 215 the differential equation of the given family of curves. If two curves cut at right angles, and if <^i, <^2) be the angles which the tangents at the intersection make with the axis of x, then <^i = <^2 ± 2> and therefore, tan <^i = — cot <^2- Hence, -^ for one curve is the same as for the other. ax dy dx Therefore, the differential equation of the system of orthogonal trajectories is obtained by substituting dx dy dy dx in equation (2). This gives Integration will give the equation in the ordinary form. Suppose that f{r, ^, c) = is the equation of the given family in polar coordinates, and that <^(n^,|)=0 (3) is the corresponding differential equation obtained by the elimina- tion of the arbitrary constant c. Let i/zj, i/fj, denote the angles which the tangents to one of the original curves and a trajectory at their point of intersection make with the radius vector to the point. Then tan i/^i = — cot \l/2- dt) Now tan 1*1 = r — Hence the differential equation of the dr required family of trajectories is obtained by substituting Idr t dO for ?-— , r de dr 216 INTEGRAL CALCULUS [Ch. XllI that is, -r^^for^ dr ae in (3). This gives fr, 6, - r'^ as the differential equation of the orthogonal system. Ex. 1. Find the orthogonal system of the family of parabolas y^ = 4: ax. Differentiating, yS. = 2 a, dx and eliminating a, y = 2x -^• dx This is the differential equation of the given family. Substitution of _dx jgj, dy dy dx dx gives y=-2x—-, dy the diiierential equation of the familyof trajectories. Integration gives y^ + 2x^ = c2, the equation of a family of ellipses whose foci are on the y-axis, and whose centers are at the origin. Ex. 2. Find the orthogonal trajectories of the cardioids r = a(l — cose). Differentiating, — = a sin 8. dd Elimination of a gives — = r cot -i de 2 the differential equation of the given family of curves. Therefore, the equa- tion of the system of trajectories is de ^e — r — = cot -• dr 2 Integration gives r = c (1 + cos 6), another system of cardioids. 108.] ORDINARY DIFFERENTIAL EQUATIONS 217 Ex. 3. Find the curve in which the perpendicular upon a tangent from the foot of the ordinate of the point of contact is constant and equal to a, determining the constant of integration in such a manner that the curve shall cut the axis of y at right angles. Ex. 4. Find the curve whose tangents cut oH intercepts from the axes the sum of which is constant. Ex. 5. Find the curve in which the perpendicular from the origin upon any tangent is of constant length a. Ex. 6. Find the curve in which the perpendicular from the origin upon the tangent is equal to the abscissa of the point of contact. Ex. 7. Find the orthogonal trajectories of the straight lines y = ex. Ex. 8. Find the curves orthogonal to the circles that touch the y-a,xis at the origin. Ex. 9. Fmd the orthogonal trajectories of the family of hyperbolas xy = k'^. Ex. 10. Find the orthogonal trajectories of the ellipses ^ + -1^ = 1, a^ a^ + \ in which \ is arbitrary. Ex. 11. Show that the system of confocal and coaxial parabolas J* = 4 a(x + a) is self-orthogonal. Ex. 12. Find the orthogonal trajectories of the system of circles r = c cos 0, which pass through the origin and have their centers on the initial line. Ex. 13. Find the orthogonal trajectories of the system of curves r" sin n8 = a". Ex. 14. Find the equation of the system of orthogonal trajectories of the family of confocal and coaxial parabolas r = -• ■' 1 + cos 9 Ex. 16. Determine the orthogonal trajectories of the system of curves )•» = a" cos nd ; and therefrom find the orthogonal trajectories of the series of lemniscates r^ = a^ cos 2 6. 218 INTEGRAL CALCULUS [Ch. XIII, Skction III. Equations of an oedeb higher than the first. 109. Equations of the form — —-fix). The solutions of equa- tions of this type can be obtained by n successive integrations. Examples have already been seen in Art. 59. Ex. 1. Solve ~ = a;2 _ 2 cos x + 3. Integrating, -5-| = | x^ — :i sin « + 3 x + Ci. Integrating, ^ = ^j x* + 2 cos x + f x^ + cjx + Cz. Integrating, S/ = ± a;^ + 2 sin x + ^* + 5l^ + C2X + C5. dx 60 2 2 Integrating, ^ = ^^ a;^ - 2 cos x + | x* + k^x^ + fex^ + C3X + C4. Ex. 2. Solve ^=/sinn«. Ex. 3. Solve — = «. Ex. 4. Solve B^- W(l-x)=0, subject to the condition that y = dy '^^'^ and x; = for x = 0. dx^~ Ex. 6. Solve ^ = x". dx" 110. Equations of the form ^Jl. = f{y). Multiplication of both members of this equation by 2-^ gives dx dx day' dx Integrating, fj^ =2jf(y)dy + Cj, 109-111.] ORDINARY DIFFERENTIAL EQUATIONS 219 whence, • ^^ r^= dx. Therefore, f , ^ ^^ -, = x + c,. Ex.1. Solve ^+a2M = 0- Multiplying by 2 ^^, 2^ -^ = -„_,m+i3„ = 0. (3) Hence, if TOj, TOj, •••, to„, are the roots of (3), the complete solu- tion of (1) is y = c^x'"' + c^x"^ + ••• + c^a;"". To a solution y = e'"'(ci + c^ + •■■ + c,_jaf~'^) of (2), there cor- responds a solution ?/ = a;"'(ci -f- Cj log a; + ••• -|-c,_i(log x)'"'') of (1), since z = log x. It can be easUy shown that the auxiliary 116.] ORDINARY DIFFERENTIAL EQUATIONS 227 equation of (2) is identical with (3). Hence, if mj is repeated r times as a root of (3), the corresponding solution of (1) is y^x'"^\ci + c.2logx+ ... +c,_i(\ogxy-^l. Ex.8. Solve x3^ + 3a:2^ + ^ + w = 0. dx^ dx'^ dx Substitution of x" for y gives (m^ + l)!" = 0, of which the roots are - 1, ^ +^^\ 1 -v^'. 2 2 Hence the solution Is y = ^ + xi ^CiCos(^logx\+ CiSm(^\ogx\\- Ex. 3. a:2 ^ + 4 X ^ + 2 a = 0. dx2 dx '^ Ex. 4. x2^-3x^ + 4w = 0. dx2 dz '^ Ex.5. x^^-3x^^+lx^y-8y = 0. dx^ dx^ dx " EXAMPLES 1. lty = Ae'" + Be-'^, prove that ^ - k'^y = 0. 2. Derive the differential equation of all circles which pass through the origin and whose centers are on the x-axis. 3. sec^xtanj/t^x + sec^t^tanxdj/ = 0. 4. xdx + ydy = a^ '-'^y-yf^ . g dx _ dy x^ — 2xy y^ — 2xy 6. (2 ax + by + g) dx +(2cy + bx + e) dy = 0. 7. (1 +x2/)!/(Zx+(l — x!/)xiii/ = 0. 8. y(2xy + e'') dx — e''dy = 0. 9. x-^-au = x + l. dx dy 10. cos2 x~^ + V = tan x. dx INTEGRAL CALC. 16 228 INTEGRAL CALCULUS [Ch. Xlli. 11. x(l-a;-^)^+(2x2_i)y = aa;3. -^ dy . „ 12. -5^ + J/ cos X = J/" sin 2 a;. 13. ^ = xV-xy- 14. p^(x + 2y) + Zp%z + y) + (y + 2x)p = 0. 15. xp'^ — 2 yp + ax = 0. 16. p2y + 2px = y. ^4.,,-0 17. e3'(p- l) + pV!' = 0. 23. ~ + 2 H'^ ,-4 + «*!/ = 0. dx* i!x- da;* da;^ rZa:- 18. 19. 20. )^=0. APPENDIX o»;o NOTE A [This note is supplementary to Art. 33] A method of decomposing a rational fraction into its partial fractions. f(x) Suppose that _, , ) ^ — ^^ is a proper rational fraction. The ^'^ F(x) (x — ay ^ ^ substitution of a for x in this fraction, {x — ay being left un- changed, and the subtraction of the fraction thus formed gives f(p) fjcd ^ fix)F(a)-f(a)F(x) . F(x)(x-ay F(a)(x-ay F{u)F(x)(x- ay ' ^^ The numerator of the fraction in the second member of (1) vanishes for x = a, and hence it is divisible by a; — a. Let the quotient be denoted by (x). Then f(x) ^ f(a) 1 (x) F(x) (x - ay F(a) (x - ay^F{a) ' F(x) (x - a)"-'' ^ -' Of the two fractions in the second member of (2) the first is one of the partial fractions required, and the second has a de- nominator of lower degree than the original fraction possesses. The second fraction can be similarly decomposed, and by the repetition of the operation all the partial fractions will be found. When the factors of the denominator are all different and of the first degree, the decomposition of a fraction can be effected very quickly. For example, on taking the fraction 229 230 INTEGRAL CALCULUS [Note A. f(x) I , / , c- , T^ and substituting a for x except in x — a, (3/ — Oij {3C — 0) \X — C) and subtracting the fraction thus formed, there is obtained /(x) /(a) _ F(x) {x — a){x — b) {x — c) (x — a) (« — b) (a — c) (x—b) {x—c)' in which F(x) is a constant or of the first degree in x. The partial fraction whose denominator is x — a, which is formed by this rule, is accordingly ^-^ The (x — a){a — b) (a — c) partial fractions whose denominators are (x — b), (x — c), can be written by symmetry. This is easily verified. For, on assuming f(^) ^ A ^ B ^ C ^ (x — a){x — b)(x—c) X — a x~b x — c and clearing of fractions, f{x) =A{x-b)(x-c) + B(x-a)(x-c) + C(x-a)(x- b). The substitution of a for x gives f{d)=A(a-b)(a-c), whence, A=- ^^ (a — b)(a — c) It will be found, on putting b for x that B^ m , (6-o)(&-c)' and on putting c for x, that C - /(c) (c — a)(c — b) Therefore, f(^) _ f(a) (x — a)(x — b)(x — c) {x — a)(a — b){a — c) + f(b) I f(c) (b - a){x -b){b-c) (c- a)(c -b){x- c) Notea-b.] appendix 231 Ex. 1. 2a;2- 1 2-2^-1 2(-3)''-l (x-2)(x + 3)(a;-5) (x - 2)(2 +3)(2 - 5) (- 3 -2)(3 + x)(- 3 - 5) 2 . 52 - 1 + (5-2)(5 + 3)(x-5) 7 17 49 15 (a; - 2) 40 (x + 3) 24 (x - 6) Ex 2 3^ + 2 3-2 + 2 ^ „ (x-2)2(x-3) (X- 2)2(2 -3) On determining i^' by subtraction, 3 x + 2 8 _ 11 (X - 2)2(x - 3) (x - 2)2 (x - 2) (x - 3) 11 11 (3-2)(x-3) (x-2)(2-3) „ 3x+2 8 , 11 11 Hence, — — —^ — — = - — + _ (X - 2)2(x - 3) (X - 2)2 x-3 X - 2 NOTE B [This note is supplementary to Art. 45] To find reduction formulae for | as™ (a + bx''^)P dao by integra- tion by parts. In what follows | x'"(a + 6a;")'' dx will be denoted by /. (a) On putting dv = x''-\a + &«")" da.-, u = a;'"-"+', it follows that v = (fl±M^^ du=(7n-n + 1) x"-" dx. nb (p + 1) Hence, j^ ^"— V + &^y-' _Z?L^^ fx-^ia + bxr^^dx. ' nb(p + l) nb{p + l)J But ar-"(a + &»")''+' = af^(a + 6af)(a + fta;")" = aar-"(a + bx")'' + 6a;'"(a + bx^y. 232 INTEGRAL CALCULUS [Note B. Therefore nb{p + l) nb{p + l)[J ' Ob solving for I, b(m + np + 1) b (m + np + l)J (b) On transposition in the result just obtained and division by a(m- 71+1) ^ b(m + n+p + l) Cx'^-'Ha + fe-x-")" da; = '«"""'''(« + &«'")'"" J a (m — w + 1) _6(m + np + l} r^.(„ + 5^..)p^. a (m — 71 + 1) J From this, on changing m into m + m, J ^ '^ a(m + l) - Mm + » + np + l) r „^„(^ ^ j^„. , ^^_ a (m + 1) J (c) On putting di) = x" dx, m = (a + 6a;")'', it follows that v = , du = »?i6a;"~Va + 6a;")''"' da; m + 1 ^ and hence, ^^^'-+V« + 6a;y_jm6_ r^™+„ („ + ;,^n),-, rf^. m + 1 m + 1 J ^ ^ But a;'»+» = x-x- = ^"'("'+''^") - ^. 6 6 On substituting this value in the last integral, j^x^(a + bx^_j>n^ri_a T^™ („ + ^,^„).-. ^; m + 1 m + 1 6 bJ ^ ' Note B-C] APPENDIX 233 Whence, on solving for /, m + np + 1 m + np + IJ On transposing in the last result and dividing by ^^ > m + 7ip + 1 From this, on changing p into p + 1, fa;"' (a + bx")" dx = - ^'^' (a + b^")"''' J ^ ^ an(p + l) + m + n + np + 1 r„ ^^ ^ , ^^_ a»i (p + 1) J KOTE C [This note is supplementary to Art. 51] To find reduction formulae for | sin™ x cos" as dx by integration by parts. On denoting this integral by /, J= j sin" a; cos" a; da; = — j sin'"~^a; cos"a;ci(cosa;). On putting dv = cos" a; d (cos x), u = sin^^'a;, it follows that v = °°^" '" , du = (m - 1) sin"*-' x cos x dx. TO + 1 Hence, j-^_ sm"'-^«^cos"+'^ + "^ - 1 fsin— a;cos»+^a;da;. But sin"-^ a; cos"+^ a; = sin" "' a; cos" a; (1 — sin^ a;) = sin"-' a; cos" a; — sin" a; cos" a;. Hence, j^_sin"--^cos"+^a; m-1 T U^^-^ ^ ^^^n ^^^ _ j\ ' n + 1 n + l[J 234 INTEGRAL CALCULUS [Note C-D. From this, on solving for I, j^_sin"-^a.cos''^^x^m-l U^^-^ ^ ^^^n ^ ^x. m + n m + n J From this result, on transposition and division by —, fsin-^x cos" X dx = si^'"-'^cos"^'«' + ^!!l±J!l fsin- x cos" x dx ; J m — 1 m — U whence, on changing m into m + 2, fsin-x cos»a. do; = ^^'^"'^' '^ "°^"^' ^ + Z?i±iL±2 f sin™+^ a; cos" x dx. J m + 1 m + 1 J Formula C, Art. 61, can be obtained by writing 7=1 cos" ■' X sin*" x d (sin a;), putting dv = sin" a; d (sin x), u = cos"" ^ x, and then integrating by parts and reducing. Formula D, Art. 51, can be derived from C by transposition and the change of n into n + 2. NOTE D [This note is supplementary to Art. 67] It is explained in the differential calcrdus that if the differ- ence between two quantities be infinitesimal compared with either of them, then the limit of their ratio is unity, and either of them can be replaced by the other in any expression involving the quantities. A deduction that can be made by means of this principle is of great importance in the practical applications of the integral calculus. If ai + ajH l-a„ represent the sum of a number of infinitesimal quantities which approaches a finite limit as n is increased indefinitely, Note D-E.] APPENDIX 236 and if p„ p„ ..., ^„ be another system of infinitesimal quantities, such that .| = l + e, 1=1 + ., .., |=l+.„, (1) where e„ e, •••, e„ are infinitesimal quantities, then the limit of the sum of ;8i, /Sj, •••, P„ is equal to the limit of the sum of Kj, a^, •••, «„. It follows from equations (1) that A+/32+ ••• +/8„=(ai+a2H |-a„) + (a^ei+a2e2+ ...+«„e„). (2) Let r] be one of the infinitesimal quantities e^, e^, •••, e„, which is not less than any one of the others. Then 08l + A + •■• + iSn) - («1 + «2 -1 h «„)<(«! + «2 H h «n) V' But by hypothesis aj + «2 + • • • + «„ has a finite limit, and hence the second member of this inequality is infinitesimal. There- fore the limit of ft + /82 H /8„ is the same as the limit of «, J-KjH [-«„•* NOTE E [This note is supplementary to Arts. 84-87] Further rules for the approximate determination of areas. A few more rules for approximately determining the area of LAKT (Fig. 63) may be stated. As before, h denotes the interval between successive equidistant ordinates, and merely the coefficients of the successive ordinates are given in the formulae. In the trapezoidal rule, strips were taken in which two ordinates were drawn; in other words, the ordinates were taken by twos. In the parabolic rule, strips were taken in which three ordinates were drawn; that is, the ordinates were taken by threes. * See B. Williamson, Treatise on the Differential Calculus, Arts. 38-40. 236 INTEGRAL CALCULUS [Note E. Rule A. If four ordinates are taken at a time, then for each, strip, approximately, area = 1^(1 + 3 + 3 + 1). This is commonly called Simpson's three-eighths rule. KuLE B. If five ordinates are taken at a time, then approxi- mately for a strip involving them, area = ^^^^(7 -f 32 -t- 12 -f 32 -f 7). Rule C. If six ordinates are taken at a time, area = iff A (if + 3 + 2 + 2 -f 3 -f if). KuLE D. If seven ordinates are taken at a time, then, after a very slight modification of the formula that first presents itself, area = ^h (1 + 6 + 1 + 6 + 1 + 5 + 1). This is known as Weddle's * rule. It may be expressed in words : The proposed area being divided into six portions by seven equidistant ordinates, to live times the sum of the even ordinates add the middle ordinate and all the odd ordinates, multiply the sum by three tenths of the common interval, and the product will be the required area approximately. Rules A and D are frequently employed. The trapezoidal and parabolic rules A, B, C, above, are special cases of one general rulef which is deduced on the supposition that the area concerned is divided into n portions bounded by n + 1 equidistant ordinates whose lengths and common distance apart are known. The given curve, say y = 4> (x), is replaced by a curve which passes through the extremities of the n + 1 given ordinates, and whose equation is a rational integral function of X of the nth degree. The area of the latter curve can be easily * It was first given by Mr. Thomas Weddle in the Cambridge and Dublin Mathematical Journal, Vol. IX. (1854), pp. 79, 80. t This rule was first given by Newton and Cotes, and published by the latter in 1722 in a tract, De Mcthodo Differentiali. Note E-P.] APPENDIX 237 found by integration. For example, in the case of each succes- sive pair of ordinates in Art. 85, the given arc was replaced by a straight line, and in the case of each successive group of three ordinates in Art. 8G, the given arc was replaced by the arc of a parabola. On assuming that the equation of the second curve is y = A^ + A^x + A^+...+A„x% (1) the coefficients A, A^, A^, •••, A„, can be determined. For, the substitution in (1) of the coordinates of the n + 1 given points, namely the extremities of the given equidistant ordinates, will give TO + 1 equations, by means of which the values of the w + 1 coefficients A^, A^, •■-, ^1„, can be found.* If n is sufficiently great, the difference between the area of the second curve and that of the original curve will generally be very small. The general formula for the case of m + 1 equidistant ordinates can also be deduced by the method of finite differences. f For a dis- cussion on various methods of finding an approximate value of a definite integral by numerical calculation, reference may be made to J. Bertrand, Calcul Integral, Chapter XII., pp. 331-352. NOTE F [This note is supplementary to Art. 88] The Fundamental Theory of the PlanimeteeJ In Fig. 58, ALBO is a plane figure whose area is required, and QX is a given straight line taken as the axis of X. Let MJSf rep- resent a plate of which two given points always move, Q along QX, and P on the contour of the given area. Then QP is a straight line fixed with reference to the instrument. Let b be the length of QP. Let W be the recording wheel with axis parallel to QP. Its actual location is arbitrary. * Also see Lamb, Infinitesimal Calcuhis, Art. 112. t See Boole, Calculus of Finite Differences, Chapter III, Arts. 10-14. t This note is by Professor W. F. Durand, who has kindly permitted its insertion here. 238 INTEGRAL CALCULUS [Note F. The movement of P from P to 0, a point very near, may be decomposed into: (1) A movement dx parallel to QX, (2) a movement dy at right angles to QX. It will first be shown that the record of the wheel W due to the dy component will, for the closed area, be zero. Fig. 58. It may be noted that the amount of the dy record depends on the dy and on the configuration of the instrument under which it is traversed. Now it is evident that for every dy traversed in the up direction, there will be an equal dy traversed in the down direction, and under the same configuration. In the diagram the pair thus traversed is dy and dy^. The net record for such a pair is zero, and for every other pair, zero, and therefore, for the entire contour, zero. It follows that the entire record will be merely that due to the (Ix components. This is found as follows. The component of dx in the direction of the plane of the wheel is dx ■ sin 6. But sin = j^. Denote that part of the record due to dx by dR. Then, dR = dx • sin $ = ydx Note F.] APPENDIX 239 Hence, and therefore B A = bB. It only remains therefore to graduate W conformably to the length of 6, or, vice versa, to graduate W and give to b an appro- priate length. The latter is the usual method. By giving to b various lengths, the area may be read off in corresponding units. Thus far it has been assumed that Q follows the straight line QX. It will next be shown that the record is independent of the path of Q so long as it is back and forth on the same line. Fio, To this end let FM (Fig. 59) be any area, and ABODE a broken line. Let A and E be the points from which arcs with a radius b will be tangent to the contour at F and M. With the same radius and B, C, D as centers, draw arcs as shown in the figure. Suppose now that P is carried around these partial areas successively, and always in the same cyclical direction. Per (1) the point Q (Fig. 58) will traverse AB, for (2), BC, eto. In each single case the record will represent the corresponding area. Therefore the total area will be represented by the total record. And it is readily seen that GH, IJ, KL, are each traversed twice in opposite directions. Hence the record due to them is zero, and the actual record is due only to the external contour. Hence, if P were carried directly around the external contour, it should 240 INTEGRAL CALCULUS [Note F-G. have the same record, and hence the area. This is true for any broken line, and hence for a curve. In the common polar planim- eter the curve is the arc of a circle. Thus the point C in Fig. 55, Art. 88, which corresponds to the point Q moving along QX in Fig. 58, or along ABODE in Fig. 59, moves along the circum- ference of the circle of center T and radius a. NOTE G On Integeal Cukvbs 1. Applications to mechanics. — (This article is supplementary to Art. 93.) (a) The statical moment about OY=My = (area) 0H= ayiOH. Hence, J/j, = ai/i (xi — HX) = ay^Xi — ahy^ = a {y^x^ — hy^). (1) Also loH = Ipz - (area) {HXf = 2 aUy, - ^^ = ab(2 cy, - ^\ (2) 2/1 V 2/i (6) The value of HX in Art. 93 (6), may be found by a simple construction, though from its nature the accuracy may not be all that is desirable. Let BH be drawn tangent to OB at B. Then tan BHX = ^= Jh_. dx HX But % = - I Vi rf'i", and hence -^ = ^■ ^ bJ-"^ ' dx b Hence, -%=^S and HX=^- HX b y. Hence, from Art. 93 (b), the point H thus determined will be the abscissa of the center of gravity as desired. (e) The moment of any area ORFH about any vertical XA is proportional to the corresponding ordinate XD of the tangent XoTE G.] APPENDIX i!41 to the second integral curve at the point E on the limiting ordi- nate HF. It has been shown in Art. 91 that ^ = -yi. From this, dx b tan Z»^X = ^^= ^^ = ^ = ^^. HK dx b ah Hence the equation to the tangent KD is y-HE = ^(x-OH); ab whence, aby = ab • HE + A(x — OH). (3) But from Art. 93, ah ■ HE is the moment of the area about HF, and A{x — OH) is the correction necessary to transfer this moment to an axis distant (a; — OH) from HF, and therefore distant x from the origin. The second member of (3) is thus seen to be equal to the moment of the area about a vertical line at any distance x from the origin. Hence, such moment is meas- ured by aby, or ab times that ordinate of the tangent line which is determined by the abscissa x. Hence such ordinate at any point bears the same relation to the moment of ORFH about the vertical line containing the ordinate, that HE does to the moment about HF. (d) It follows that where KD crosses OX, the moment about the corresponding ordinate will be zero, and hence an ordinate through /^will contain the center of gravity of the area ORFH. Hence the construction given in (h) above is a special case of (c). (e) If we apply the same proposition to the moments of the two areas ORFH and ORPX about an ordinate through L, the point of intersection of the two tangents at E and B, we shall have for each moment the expressions ohNL, and the moment of the difference of the two areas or of HFPX about ^*S' will be zero, and therefore NS will contain the center of gravity of such area. 242 INTEGRAL CALCULUS [Note G. Hence the tangents to the second integral curve at any two ordinates intersect on the ordinate which contains the center of gravity of the area of the fundamental curve lying between the two ordinates chosen. 2. Applications in engineering and in electricity. The limits of the present article will not allow detailed reference to the various ways in which these curves may be made of use in studying engineering problems. A few brief references may, however, be made to some of the more common applications. It is readily seen by comparison with text-books on mechanics that if for the fundamental we take the curve of net external force on a beam or girder, then the first integral of such funda- mental will give the entire history of the shear from one end to the other. Also that the second integral will give similarly the entire history of the bending moment from one end to the other. This serves to illustrate one important advantage of repiresenta- tion bj' means of these curves, and that is, that they serve to give not only the value at some one or more desired jjoints, but at all points as well. In this way they furnish a continuous history of the variation of the function in question, and thus give a far more vivid picture of its characteristics than can be obtained in any other way. In the case of beams or girders it may be well to note that external forces should not be assumed as concentrated at a point, but should rather be considered as distributed over a length equal to that occupied by the object to the existence of which they are due. Thus the supporting forces at the ends of a bridge span must not be considered as located at a point, as is common in the analytical treatment, but rather as distributed over a length equal to that occupied by the supporting pier. Their graphical representation will therefore be a rectangle, or rather it may be so taken for all practical purposes. As another application, consider the action of a varying efEort or force acting through moving parts having inertia, and upon Note G.] APPENDIX 2i3 a dissimilarly varying resistance, their mean values being of course the same. This is the case with the ordinary steam engine or other prime mover operating against a variable resist- ance. Suppose that we have plotted on a distance abscissa, the curves of effort and of resistance. The integral of the first will give the history of the work as done by the agent or effort, while that of the second will give the history of the work as done upon the resistance. Steady conditions being assumed, their mean values will be the same. Their history, however, will be quite different. The difference between the ordinates at any point will give the work stored as energy in the moving parts during their accelera- tion when the effort is greater than the resistance, restored during their retardation when the effort is less than the resist- ance. We might reach the same results by taking as our funda- mental the difference between the curves of effort and resistance. The integral of this will give the history of the ebb and flow of energy from and into the moving parts of the mechanism. Again, by replotting this latter curve on a time abscissa it becomes representa;tive of the time history of the acceleration of the moving parts. If then the reduced inertia of these parts is known, the acceleration at any instant is known, and the curve may be considered as one of acceleration. Its integral will, therefore, give velocity, such velocity being the increase or decrease above the mean value. Such a curve would, therefore, show the continuous history of variation in the velocity due to the causes mentioned. In electrical science there are many interesting applications of these methods. Of these only one or two of the simpler will be here given. Suppose that we have on a time abscissa a curve showing the history of the electromotive force in any circuit. Then since this is the time rate of variation of the total magnetism in the circuit, it is evident that, reciprocally, the latter must be the INTEGRAL OALC. 17 244 INTEGRAL CALCULUS [Note G. integral of the former. Hence the first integral curve will give the history of the total magnetic flux in the circuit. Again, if we have on a time abscissa a curve showing the history of a current, then the history of the growth of the quan- tity of electricity will be given by the first integral of such curve. Instances might be widely multiplied, but enough has been given to show that where desired results may be found by one or more integrations effected on a function whose history is known, the complete representation of the problem naturally leads to the production of these curves ; and for their practical determination and for their application to many special problems, the fundamental relations and properties as developed above and in Chapter XII., will be found of considerable value. 3. The theory of the integraph. We will next show briefly the fundamental theory of the integraph, an instrument for practi- cally drawing the first integral from its fundamental. Various forms of instrument have been devised, but in nearly all, the kinematic conditions to be fulfilled are the same. These are as Note G.] APPEJ^DIX 245 follows : Let PC (Pig. 60) be the fundamental relative to axes OX, OY; and QD the integral relative to axes QX„ QY. For convenience the two F-axes are taken in the same line, though this is not necessary. P and Q are therefore corresponding points. At Q draw a line QA tangent to QD. From P draw PE parallel to QA. Then : OA dx OE a Hence, -^^ = ^ or w, = \-dx. ' dx a "^ J a If now a is constant, we shall have 1 C J area Wi = - \yax = or area = ay-y. aJ a These conditions are seen to correspond to (2), Art. 91. The instrument must therefore include three points, E, P, and Q, related as above specified. While the instrument travels along the direction of X, P is made to trace the given curve, and E remains at a constant distance a from the foot of the ordinate through P. This determines a direction EP, and Q constrained by the structure of the instrument to move always parallel to EP, will trace the integral curve QD. It is not necessary that the points E and A should lie to the left of as in Fig. 60. They may be taken as at E', A', and in such case if the fundamental lies above X, the integral will lie below its X as shown by QD', and vice versa. The actual values, however, will remain unchanged, and the inversion is readily allowed for in the interpretation of the results. Following are the figures of some of the curves referred to in the preceding pages : The hypocycloid, x'' + y'^ = aJ. The cissoid, y^ = 2a — X O O' X The cycloid, x = a{e — sin 6) ; y = a(l — cosS). 246 Folium of Descartes, x^ + y^ = Zaxy. X X The catenary, y = - (e" 4- e «). 2 The parabola, x^ + y- -= a/-. The semicubical parabola, ay^ = x?. The cubical parabola, a^y = a;8. The parabola, r = a sec^ — The logarithmic spiral, r = e"*, 247 The curve, ?• = a sin» — o Spiral of Archimedes, r = aS. The curve, r = a siii2 tf. The cardioid, r = a(l — cos 'i"he lemniscate, r' = a^ cos 2 d. The witch, y -. ■ x2 + 4 a^" 248 A SHORT TABLE OF INTEGRALS oXKo Following is a list of integrals for reference in the solution of practical problems. The deduction of these integrals will be a useful exercise in the review of the earlier part of the book. GENERAL FORMULA OF INTEGRATION 1. j (u ±v ±w ± ■■■)dx= I udx ± I vdx ± i w dx ± •■•. 2. I mu dx = m I u dx. 3 (a). I udv = uv — I V du. 3 (b). i u — dx = uv — \v — dx. J dx J dx ALGEBRAIC FORMS 4. I — = log X. J X x" dx = , when n is different from — 1. n + 1 Expressions containing Integral Powers of a + 6a; ^ r-^=^iog(«+to). J a + bx h 7. C(a + bxY dx = ("« + ^'g)"""' , ^hen n is different from - 1. J ' 6 (n + 1) 249 250 INTEGRAL CALCULUS 8. I F{x, a + bx) dx. Try one of the substitutions, z = a + bx, xz = a + bx. 10. I '-^-^ = 1 [1 (a + bxf -2a{a + bx)+a' log (a + ta;)]. 11. f ^^-^^ =_lloggdl&^. »/ a; (a + 6a;) a a; 12. ./ ar'(a das 1,6, a + bx , , , s = l-^log— != (a + 6a;) aa; a"^ a; 13. f-^ J (an .I'da; log(a + 6a;) + a ~ | a + 6a;J (a + 6a;)2 6' ^*- ^ I "It ^^ = f, fa + 6a; - 2 a log (a + 6a;) f-l J (a -t- 6.1-)'' 6''|_ a+6a;J 5. r '^■'- = 1 iiog^^+^. J xia-V- bxY « (a + 6a;) a^ a; 15 16 (a + 6a;)^ « (a + 6a;) r X dx ^ 1 r 1 J (a + 6a;)'=~6= - + a + 6a; 2 (a + 6a;) ^ Expressions containing a^ + a;^, a' — os', a + bx?', a-\-bxP 17. r-^ = ltan-^; r^ = tan-,. i/a. +ar a a J l+ar 18- r-i^ = f log^±^; r^ = ^log^. ^ a"^ — ar z o a — x J or — a 2 a x + a 19. I ;= tan"'a;\/-, wheii a>0 and 6>0. J a + hx^ -^f^ Va J a2 _ 7,2.,,2 20 1 1 a + 6a; -log- ^ - Wa? 2ab a — bx A SHORT TABLE OF INTEGRALS 251 21. Caf'{a + b3f)''dx b(np + m + r) b{np + m + T)J ^ ' 22. iLyr{a-\-bafydx ^ ^^\a^b^y ^ anp T „ ^^ _^ 6a,")->dx. wp + m + 1 np + m + 1«/ 23. f— r-*^"" J af (a 24 J a;"" (a (a +60;")'' 1 (m—n+np—l)b f dx (m— l)aa;'" ^{a+bx"y-^ (m— l)a J a;"'-"(a+6a;")'' dx (a + 6af )" ^ 1 an (p — 1) a;"-! (a + da;")*-' ' a7i{p — l) J ar{a + bx")"'^ 25 r (a + bafydx a(m — l)ar~' a(m — 1) 1 . m — w + wp — 1 r dx ry-' a7t(io — 1) J a; _ _ (g. + 6ai")''+^ _ b(m — n — np — l) r (a + 6a;"y da; "~ J a;"*-" 26. f (a + &a;">'da; _ (ft + bx'^y anp r ( ~ (np — m + l)af'^ np — m+lJ X'" 27. f-^^^ -' (a + (a + bx")' »»■-" + ' a(m — w + 1) C 3r-''dx bim-np + ljia + bx")"-^ b(m,-np + l} / a;"~"da; (ft + 6a;")'' 28. r , •'^"^,'" . J (o + 6a;" (o + 6a;"y a;"'+^ m + n — np + 1 T a;" da; ' o«(p - 1) (ft + bx")'-'^ an (p-1) J (ft + baf^y- 252 29 I J (oF + af)" INTEGRAL CALCULUS 2(w-l)a' dx X _(a^ + oiF) -^+(^'^-^)/k5p-] 30. r — '^ — J (a + 6a;^)» = - 1 r "^ + (2n - 3) f ^ 1. 3j_ r xdx ^ir dz i.ere.^ar'. J (a + 6x^)" 2 J (a + 6»)" 32. C ^^^^ T" — g; I 1 /* dx ~ 2h{n-l) {a + bx^"-'' 2h{n -1)J (a + Sar^""*' da- ^ 1 J r (a + 6a;") an 34. 35. p 33. C—l^— = Uog-^—. J x{a-\- bx") an a + baf ^ 'j:\^ -Y ox ) an a -\- ujx . r dx _1 r dx 6 r dx J x'{a + bxy ~ aJ x\a+ bxy- ^ a J (a + &af)' c r xdx In / o , a\ a + fea;'' 26 ^^ 6 da; „-, /* a.'^dg _ 5 _ ^ r__dx__ J a + by? b bJ a + ba? 37. C '^ = — log — - — J x(a + bx^ 2 a a + ba? „Q r dx _ _ J^ 6 r dx J 3?(a + bv?) ax a J a + bv? „q r dx _ X _l_ r dx J {a + bxy~'2a{a + ba?) YaJ a + bx^' A SHOBT TABLE OF INTEGRALS 258 EXPKBSSIONS CONTAINING ■\/a + hX [See Formulse 21-28] 40. f«.V^+65dx = -2i2a-3MV(a+M!. J 15 W J 105 b^ 42. r_4^^=_2i2az=Mv^Tto. 43 r a^t^a: ^ 2 (8 a^ - 4 aSa; + 3 S^a;^ ^/^|^^_ ' -^ y/a + bx 15W 44. f g^ = -l_log -^" + ^'"-^ . for a>0. 45. ( ^^ ^^_tan-'-J^+^, fora<0. •^ a; Va + 6a; V— a * — a • „ /^ da; _ — Vo-Fft^ _ ^ C c^a? / da; _ — V g + 5a; 6_ T d a?Va + bx aa' 2aJa!Va+&a5 47. f^^«+M^=2V^+te + aY— ^^-. J X J a; Va + 6a; Expressions containing Va!^ + a* [See Formulse 21-28J 48. r(a;2^^2)i^a.^|y^qr^2 4.^1og(a, + V^+<). 49. r(a;^+a^* da; = | (2 oj^^ 5 a2)A^T^ + ?|^log (a!+ a^T^^). 50. C{?? + a^^dx = <'^+f>\ -% C{^ + a?f-'dx. J n + 1 n + lJ 254 INTEGRAL CALCULUS 51. jx{a? + aFfdx = ^^^^^^. 52. rx^(c(^+a^*da; = |(2a^ + a^V?+¥2-|*log(x+Va^+a^. 53 ■ i (a^ + ay 54. f- '^'^ /a; da; (»2 + a^)* 55. '"«'" =V^^+^^. 56. f^!^ = ^VS^T^^_|log(a; + V?T^O. 57. f_^^ = -^— + logfa+V^+^^ 58. r ^^ ^ llog^" jCa^^ + a")* ** a+Va^ + a' x'ix' + a^^ «'^ 60 r ^. '^ (a^ - w^^J 2 2 a ow r K^da; a; ■ .a; 87. I ; = — - — sin^— '■/: (a2 _ ^)l Va^ - x^ A SHORT TABLE OF INTEGRALS 257 88. C '""^'^ =_g::^v^^z:^+ ("^-i)«Y ''""' dx. 89. f ^^ — 7 = ilog "^ a? {a?— 3?)^ «'= qj, r dx Va^ — a;^ '^ If 91. f ^ = _V^!Z^ + J_log "^ nn r(«^ — »^)*j /-2 — Z& 1 a+Va' — a 92. I '^ — dx = Va^ — ar — a log J a; ° X 93. r(^5!^^d«,==_^^«!E?_siu-i J a? X a Expressions containing V2 aa; — as^, V2 aa; + a^ [See Formulae 21-28] 94. rV2 aa; - a^ da; = ^^ V2 ax - a;^ ^ «' .^^ers"' -■ J 2 2 a 95. r '^^ =Yers-^g; f-^^- = yers"^ a;. «/ V2 aa; - K^ « *^ V2a;-a^ r /K 5, a;"'~'(2aa; — a;^)^ 96. I a;'»v2aa! — ar*da; = ^ — — ^ — — (2m + l)a r ^^^^ ^ m + 2 J 97. f- J X dx _ V2 ax — a? ^■\j2ax-^ " (2 m- 1) ax™ m — 1 r dx ) a J a."-! (2 m - 1) a J a;"-! V2 ax - x^ l'58 integral calculus 98. / ardx _ _ x" W2ax~a? {2 m — 1) a C a!"-'dx ■y/2ax-a? ™ m ^ V2 ax — of 00. rV2aa;-a;^ , ^ {2ax~x^)i m-3 rV2ax~x' ,_ J x" '^ (2m-3)ax"' (am-3)aJ a;"-' *' /' /?5 --J 3a^+aa;— 2a;^ /;5 -^ , a' _, a; a;v2aa;— araa;= ^— V2aa;— ar-J — vers -• 6 2 a ' V2 aa; — a^ *^ 01 C ^^'^' — ^^ aie — af •^ ^ V2 aa; — a^ 02. f ^^^ =-V2aa;-a;^ + avers-'-- -^ V2 ax - a^ a 03. I — z=^^^^ = h^ — V2 ax — x^ + -a^ vers ' — 04. j ^^5^- — dx = -\/2 ax — a? + a vers"' — J X a 05. r:v^2^£E^dx = -2V2«^EZ_vers-'?. i/ a;- X a „„ / 'V2 aa; — or ; _ _ (2 aa; — x^i ■ J ^ '■"" 3aa^ «y r da; a; — a 08. (2aa;-a^l «V2aa;-«2 xdx X •^ (2ax- af)f « V2 aa; - ar' j J?'(a;, V2aa;— a;^ c?a;= j J'(z+a, y/a^-z^ dz, where «=a;— a. I. r '^•'" =log(a;4-a4-V2aa; + ar^. *^ V'2aa; + a^ 11. j i?'(a;, V2aa;+a;^)dx= j F(z—a, -Vz^—a'')dz, where «=a;+a A sbout tasle of integrals 259 Expressions containing a + bx ±ca^ 112. f--^— 5 = ^ tan- _2^^±&_, ^hen b^ < 4 ac. J a + bx + cx" V4ac-62 V4 ac - 6^ 113. = ^ log 2cx + b-^b^-i ac^ ^^^^ y/b^ - 4 ac 2ca; + 6 + V6^ - 4ac 6^ > 4 ac. ^ 1 : + bx-ca^ V6^ + 4 ac '"" VF+4^c - 2 co; + &' 114, r <^^ ^ 1 log V6^ + 4ac + 2ca;-6 J a + 6a; — ca^ ^/h^ 4- a ar. '. 115. r '^^ — = ^log(2ca; + ft + 2V;;Va+ft3; + c3;^. *^ Vo + te + ca? Vc 116. I Va + 6a; + ca^^da; =.^-^±^Va+6a;+cx^- ^'~^"° log(2 ca;+6+2V^V^+6i+^). 4c 8c' 117. r ^^ ^-J-sin-' 2ca;-_6_ _ ♦^ Va + 6x — cx^ Vc V6^ + 4ac 118. j Va + te — ca" da; 2 ca; — 6 / — n^ -^ , &^ + 4 ac ■ _, 2 ca; — 6 = Va + te — carH sin ' — » 4c ggf V&' + 4ac / ^ a; da; •^ Va + &a; + ca;^ ^Va±bx±c^ b_ j^g ^2 ca; + 6 + 2V3Va + bx + caf). 119 120 I a; da; _ Va + 6a; — ca;^ b ■ i 2 ca; — 6 '/: Va+6a; — ca;^ c 2 c^ V6^+ 4 ac Other Algebraic Expressions 121. rj^+^da, = V(a+a;)(6+a;) + (a-6)log(Va+S+V6+^). •^ » 6 +3/ 122. fJ^^ dx = V(a-a;)(6 + a;) + (a + 6) sin-^J^±|. INTEGRAL CALC. — 18 260 INTEGRAL CALCULUS 123. Cyj^^dx = - V(a + a;)(6-a;) -(a + b) sin-^-yj- 1 + 6 124. ^ ' ' " 1. r^^-t^ dx = - Vr^^^ + sin-' ». 125. r '^'^ = 2 sin-' \/^^^- •^ V(a; - a)(l3 - x) ^ 13 - a EXPONENTIAL AND TRIGONOMETRIC EXPRESSIONS 126. Ca'dx = -^- 128. Ce-'dx = —- J log a J a 127. (e'dx = e'. 129. | sin a; da; = — cos a;. 130. I cos a; da; = sin a;. 131. I tan X dx = log sec a; = — log cos x. 132. j cot a; da; = log sin a;. 133. fsec a; da; = f-i^ = log (sec x + tan a;) = log tan f'^ + - J J cos a; \4 2 //* da; / X cosec a; da; = I = log (cosec x — cot a;) = log tan - J sm a; ^ ^ ^ 2 135. I sec^ a; da; = tan a;. 136. j cosec^a;da5 = — cota;. 137. I sec X tan a; da; = sec x. 138. J eoseca;cot a; da; = — cosec x. 139. fsin^ a; da; = ^ - i sin 2 «. X 1 A SHORT TABLE OF INTEGRALS 261 140. fcos^ a; da; = - + - sin 2 iB. J 2 4 1/11 C ■ n J sin""^ X cos X , n — 1 C ■ „ i j 141. I sin"a;da; = 1 i sm""^ a; t^a;. J n n J ^^n C n j cos"~^a; sina; , n — 1 C >.-2 j 142. I cos" a; da; = ■-{ I cos" ^a;da;. J n n J 143 C dx _ 1 cos X TO — 2 C dx J sin" a; to — 1 sin""' a; m — 1 J sin"~^a; 1 4.4. C dx _ 1 sin x n — 2 T dx J cos" a; TO — 1 COS""' a; n — 1 J cos"~^a; 145. rcos"'a;sin"a;da; = °°^'""''" sin"+'a; _^ m--l rcos"'-^a;sin"a;da;. J m + n m + nJ 146. I cos" a; sin" a; da; .sin"~'a;cos"'+-'a; , n — 1 C m • »-2 j — H I cos^ajsm" ''a; da;. m + nJ 147. f— J sii m + »i, m + n^ dx sin"'a;cos"a; J 1 m + TO — 2 r d» s"~^a; n — 1 J i TO — 1 sin"'"'a;cos"~^a; to — 1 J sin" a; cos"~^ a; da; 148. f^ t/ sii ,m + n — 2r dx m — 1 sin"~' X cos"~' x m — 1 J sin"~^ x cos" x sm" x cos" X 1 1 , m + TO — 2 1 4Q /^cos" a; da; cos^+'a; _m — n + 2 Tcos" a; da; ^ sin" a; (w — 1) sin""' a; w — 1 J sin"~^a; jtn /^ cos" a; da; __ cos""^a; m — 1 r cos""'a;da; J sin"a; (m — w)sin""'a; wi — to J sin"a; C0S"+' X 151. j sin a; cos" a; da; = -^ J TO +1 1 Ko /^ • n J sin"+^ a; 153. I sin"a; cos a; da; = -— J n+1 262 INTEGRAL CALCULUS 153. Ctw.'^xdx = ^^^^^-Ci&-n^-''xdx. 164. Cco\rxdx = - S^^^-j'cot"-'xdx. -.. C ■ ■ 1 sin (m + n)x , sin (m — n) x 155. I sm mx sm nx dx = -7 ^ — I — pry r — J 2{m + n) 2(w-n) -.„ r , sin (m + n)x , sin (mi — n) x 156. I cos mx cos nx dx = — --) ^ — | — --7 ^ • J 2(m + n) 2 (m — n) ,_„/". , cos(m + ?i)a; cos(m — ?i.)a; 157. I sm mx cos nx dx = —f ^ -7 (— • J 2(m +n) 2 (m — n) 158. r — = ^ tan-^ (^x/^^ tan ^\ when a>b. J a + bcos X -^'a? — b^ \^a-\-b 2) V& — a tan - + Vfe + a 159. = - log , wliena<6. V&'-a' V&^tan|-V&+a atan- + & 160. f — = ^ tan-' — when a > &. Ja + &sina; Va' - 6^ Va'-6^ a tan - + & — V6^— a^ 161. = -log ^^ , when a < 6. V?''-«'-' rt tan ^' + 6 + V6^^^' 162. r ^^ =ltan->^^t?5^Y ./ a^ cos^ x + 6^ sni- x ab \ a J inn r „r ■ 7 e°'( a sin )?.!; — n cos na;) 163. I e"sm nx dx = — ^ ~ ^ : J a- + n^ /-, e' (sin X — cos x) e' sm xdx= — ^ ^— ,„. r „, , e'"'(?i,sinna' + rtcos«a;) 164. I e" cos nx dx = — ^^ — '—- i ; /, 6°" (sin a; + cos a;) e' cos X dx = — 5^ — ! '■• ANSWERS TO THE EXAMPLES CHAPTER II Art. 12 3. y = ce'^ ; y<' = e'. 4. y^ = k{x^ + c^). CHAPTER III Art. 18 "5 ' TT' m + K + l' lOo' 22 ' 38 ' _8_ j^ _j_^ _j_^ -J_, L_, L_. ■ 225' 40' 6a^' Qx"' mx™' 99 x^^' ■ 20x^0 6. |xt, t«^ fx^, ^_.^^ |x^ ix* ^_x-^^ P+ q 4«c. V62-4ao 2ax+b+ Vb^ - 4 ac 43. —log (2 ax + 6 + 2VaVax^ + bx + c). Va 44. — sin-i ^"^-^ ■ 46. sinxcos2/ + c. Va ■\/b^~+iac 45. — cos X cos 2/ + c. 47. x^ + 3x'^y+ixy^ + 2y^ + c. CHAPTER IV Art. 29 1. («) ¥; (&) 4. 6. i. 11- 2. 2. (a) 74|; (&) f. 7. 25|. 12- i'r'^- 3. (a) tV; (6)4. 8. i^im. 13. |. 4. 9|. 9. 2|. ^ , 5. 2811. 10. 24. "■ ^^1°^!- Art. 30 5. Sl^T- 6. (a) ifa^; (6) ^^; (c) T^,r; iffir. 7. (o) fTr; (6) 2,r; (c) ^^x; (d) V-»- 8. (a) 4t; (6) TrosMogCH- V2). 9. |,rc*xii 10. ^c^xii 11. ^. 12. ixo6=. 7 / C 268 ANSWERS Art. 32 3. Vn y = X + c; Vn (y — 3) = x — 2. i. y = ce". 5. (n + l)y^ = 2 }cx"+^ + o. 6. cr = e". 7. )■" = c sin nS ; )• = c sin 9 ; )• = c (1 — cos 8). Page 76 1 _il 2f 60 63' 2. 4f. 3. -i^a2. 6. log4-|. 7. («)<^; (6)1^. 12 12 8. i. 9. 4 a". 10. 11. 2 a-. 18. ?{-!-)• la. 13. 32 . h^ 3 . 5 . 7 . a^' 19. 20. 2 TT^a^. 14. fa6. 21. ^T. 15. («) H'^; (6) irH^. 23. 350 7r2. 17. 15' CHAPTER V Art. 34 7. (X- a)' by 26. \3 2i 2 W^-'^ 7 in^ C^-"!)' 13. x2+log(x+l)S(x-4). x + 2 3. log(x+5)^(x-7)'. 8. log[V2^3T(x+2)].l*- | + 5x-logx3(x+2). 4. -log(x-2)2(x-l). ^- il0S«(«'-3)«. (x_p)(x + 5) 10. log(x + 3)2(x-2). 15- log ■ 5. log^^^iEI. ^^ log^. ^ 2 * ^l-x2 16. x + log^^^. 6. log?^^. 12. log ^-^-A » X - 2 + V3 17. log Vx^ + 2 X - 4. 18. TVlog»(-c - 1)''(^ - 2)-»(x - 3)9. io 1 i„„3;— -\/2 , 1 1 x—VS 19. log H :; log 2V2 X + V2 2\/3 X + V3 20. i.log(x2- 1)- }logx(x2-2)+ jVlog(«2-4). -, (a~a)(a-h)(a-c) . . , (^ - a) (^ - 6)(/3 - c) , . „, C7-a)(7-« ^^ ^^- AKS WERS 269 Art. 35 2. log(a;+l) + ^— . 9. alog(a; + a) + -^'*' 3. log(x-3) ^. 10. logx(x-iy--- x — 3 X 1 11 7 , 11 ,„„a;+ 1 4. logV2a; + l -2 i 11. ! + — log 2 2a; + l 21 + 2 4 ^ a; + 3 5. log(a: + a)«(a;+6)-' ^. 12. -1--+ log(a; + 1). X + X + 2 6. 1 13. 5 + ^_iog^+^. (.3V5-2-a;)2 4(x2 - 2) 8V2 x-V2 7. log(.-3)3-A^ + ,^l^. 14. l^z^, + ,l^iL + log? ■ x-3 2(a;-3)2 (x - 2)2 2(x+l)2 x + 1 8. log^±l-^. 15. -^^ + logi^^iIl X X + 1 ix-iy X Art. 36 3. logx + 2taii-ix. 6. ten-ix+f — ® -J (X2 + 1)2 4. log(x + l)2 + tan-ix. y, ilog(3.>; + 2)- i tan-i (x + 1). 5. ^+iiogx + — tan-i^-. 8. 31ogx+ — tan-i^^^. 3 ^ ^ 3 V2 V2 V2 9. ^_I^+5x. 3 2 V3 V3 •'(:^^ + 3)2 J(x2 + 3)3 V5 V5 6 (a;2 + 5)2 12. x + llog5^-V3tan-i^. 6 " x2 V3 13. log Ji^±^ + a tan-i ^ - A tan--^. Vx2+2 " 2 V2 v^ CHAPTER VI Art. 38 a: 4. ^ 6. -V^l+«! «Va2-x2 ' «Vx2 + a2 ' a2x Va2-x2 ^ (x2-a2)l 3. - * a^y-j-a _ (j2 a^x 3a2x3 270 ANSWERS 8. -llog( " + ^«^+^ V 9. -hoo f +^«^^ ). 11. - ^2 ax — a;2 12. 1 . , a - - sill"! — a X -(2 ax -x^y _ Sax^ «V'2 ax - a;2 13. Art. 40 3. 3 xU J logfcil' _ V3 tan-i f ^^i + ^N . ' x-1 V V3 i 4. x'f + log(a;^ + l). a. up-f .f -f f ). 3 2 J _ 1. Ji _ i 1 '\^~I' 7 ' 3 ' 5 4 3 e, i2/'d_^i_^+^ + »^-_«I-^+x*+xTV_log(x*+l)-tan- ■ia;T^y 7. I2f '^ - L^ + ?i + i^ + xtV + log(xT J - l)S(xT^ + l)2y V 5 4 .) 2 y 8. ^^(a+ 6x)^(3to-2ry). 1 5 h^ 9. log(3 + 2V3; + 1). 10. X + 1 + i\'x + 1 + 4log ( Vx +1-1). 11. A(o-x)*(5x + ac-24). 12. 3 2x + I (2x + 3)^ Art. 41 3. - x2 + 2 Vl - x2. 4. -J-loJ^^^^^^--^^ 2 V:i V Vl - X- + \/2 -^ 5. Vx2T5--^lo] V3 1 ,„„.^^/S^^T5-y3 ° Wx^ + 5+\/3/ i + 5+\/3/ Art. 43 2. _ J-tan-if^^^^i^^V 3. — log(3 + 4x + 2 V2v'2x^ + 3x + 4). V2 \ V2x / V2 4. log (x 4- 1 + V2 X + X-) - X + \/2 X + x'^ 5. Vx2 + X + i + Jlog (x + J + Vx2 + X + 1). Vl - x2 ,,„„/vT^^ + 6. 2x2 -ilog (vai). ,._,v¥+ »-(¥)■ AN S WEBS 271 Art. 44 2. \/51og(5a;-l+V5\/5x2-2x + 7). 3. A:log(6a;-l + 2V3V3x2-a;+l). V3 4. sin-i^^Jzjy 5. 2V2sin-'[ ^'"~'^ Y 6. _|V6-3x-2x-^ + ^^sm-ifi^V 4V2 V Vs? / 7. a sin-i - - Va^ - x^. 8. 5 Vx^ - 3 + log (x + Vx^ - 3) . a 13 9. |V3x2-3x + l + -^log(6x-3 + 2V3v'3x2-3x + l). 2V3 33 12. i V5x-^ - 26x + 34 + -^^log (5x- 13 + V5 VSx^ -26x + 34) 5V5 + 13 log ( 2x-7 + V5x^^-26.x + 34 \ 13. lpg/ 2 + x + 2Vx^ + X:H \ 14. 2sm-i(^^) - 4 V21og ( ^ + ^^1_+ ^ ^^^). Art. 45 X 4. -V2ax-x2 + aTers-i-- 5. - 1 Va^ _ a;2 + ^ gin-i a 2 2 a g .. ^ Vn^^^2 1 ,_^„ + VS2-r^'^ a^vV^^ 7. -^"'-^"-^logf^ 2aV 2a3 °\ 8. _ i(,3 «2 + ax - 2 x2) V2 ax - x2 + — vers-i -• 2 o 3a*x^ Page 98 1. log(2x + a + 2Vx2 + ax). ' 2. log (2x - 5 + 2Vx2 _ 5x + 6). ilf, 3. —log [2 NH - P+2 NVNH^ - Fx + B^. 4. log (X - a) (2 X - ft - 6 + 2 \/(x - a)(x - 6)). 5. -i-log(4x3 + 3 + 2v^V2x'i + 3x3 + 1). 3v^ 272 ANSWERS 6. isin-iV3 (a; + 1). 7. — log[V5(a; + 1) + V5a;2 + lOx - 27]. 8. ^sin-™'^ h X — 1 Vn « 17 17 ' ' 17 V2x-x2 10. |log(2x + V4a:2-9) + ^sia-' / I^x-IS N V6 V 5 / 11. - 2 Vx2 + 2x + 4 + 6 1og(x + l+ Vx2 + 2 X + 4). 12. - i (2x+ llV(x - 2)(3 - x) + -V5-sm-i(2x - 5) 13. \ (X - 4) Vx'^ - 1 + I log (x + Vx2 - 1). )■ / V^ Vx^ + 4x + 5 — X — 1 16. log(x + 2+Vx2 + 4x + 5) + — ^logl V2 \ X + 3 17. logT x + l+Vl+2x-x^ )-X \/2 + Vl + 2 X — x' X- 1 )• 18. tan-i\/x2-3. 19. Vx^ + I + 21og(x + VS^Tr)-^log('l-^ + ^^^^^ + ~l^ V2 V x + Va-^+ 1\ 20. ? ''I — X + sin"i X H sin 4\/2 , /SxJ^V U + 3 / 6*1 21. ^x* + a&2x2 + ^Vfi-'' + a2x2(3 J-^x + 2 aH^) + ^ log ( VPTo^ + ax). 22 xC3a^-2x') 3a*(a2_x2)-' 23. 48 24. - Va2 - 'r% Va^ - x-^(8 X* + 10 a%;2 + 15 a*) + ^ sin- ^sin-i?. 16 x:'. 26. - tV ^«^ - x2 (3 x» + 4 a%2 + 8 ««) . li„„/«+Va^-»^\. 27. V a^ ^ ^2 _ „ i„„ /« + Va'^ - x2^ 26. -llog( "+^°^-^^ V 28. - Va-^ - x2 (2 x2 - a2) + ^sin-i- :r2_aiog( « + v^«-^-^'' V 29. - Va2 _ x2 (8x* - 2 0^x2 - 3 a*) + -^ sin-i 48 30. ^Vx^±«?. 31. -ilogf a^x a \. 32. - Vx2 ± a- (2 x2 ± a2) - ~ log (x + Vx^ ± a^). 1, „ /a + \/xM^\ 33. ±x a'^Vx^ ± a^ ANSWERS 273 34. I Va;2 ± a2 (2 a;2 ± 5 a^) + ^ log {x + Vx' ± a'). 35. Vx'' ± a'. o 8 36. -■v/2ax-a;2 + avers-i-. 37. _ ^^ ax - g-^ _ a aa; 3g_ _ 2a;^ + 5a (X + 3a) ^ ^ ^ _ ^, 5a^ ^^^^_, x_ 6 2 a 39. ^^^V2ax-x-^ + «'vers-i5. 40. ^ii^L+l^ + _^ tan-i *■ 2 2 o 8 at (o2 + x2)2 8 a° a 41. ^ + JLtan-i^. 42. ^ 4a*(a^ + a*) 4a° a^ 3 (a^ - x^) 43. ^^^^^ + ^_tan-i f^^^V 44. 2 + Uog 4(x2-2x + 3) 4V2 V V2 / 2(l-x-'') x+1 X- l' 45. Vl + x+x^-^log(2x+l+2v'l + g+x--')-log( ^ x+2\/l+ x+x^ \ \ x + 1 / 46. - tan-i \/x + 2 V2 tan-iA/- - Vi? tan-'J^- 47 gi„-, 2 a^6^ - (a^ + 6^) (a^ + 6'^ - x^) , ^g t^"!. (a2-62)(a2-|-62._3;2-) ■ 4 49. ^^-^ ■ 50. ^Vx^ + 2 ^ 4 log (-^ ^ Vx''' + 2). 2 (x^ + 3) 2 jj_ 2 « + 5 .^4 ^2 ^ 4 ^ ^ 3 ^ , log (-2 a; _|_ 1 _)_ V4 X- + 4 X + 3) . 16 x-1. 52. ^-^^Vx2 + 2x + 3. 53. - — V- 3 + 12x - 9x2 -ff sin-i (3x-2). 2 18 54. ±(2;x + »r»)V ;a'^ + „^ + n + '"'-^^'' sin-i( ^J^ + iJ^ V 4^ 8?V-Z Wm2-4Zn/ 1 r (x + Vx^ - a^)^ + (3 - 2 V2) a^ 1 ■ 2aV2 ^ L(x + Vx-i - 0-^)2 + (3 + 2 V2) a^J .„ 2V3x^- 2x~+~l 66. ; 2x-l CHAPTER VII Art. 46 5. (a) sin x — sin^ x + | sin^ » — 7 sin' x ; (6) — cos X + f cos^ X — i cos^ X ; (c) — cos X + cosS X — I cos* x+ \ cos ' X. 274 ANSWERS 6. (a) - -c°^^^'°^ (sin-^a:+|) + |a;. 4 (6) - ^ sin5 X cos X + | ("sin* ziZx, [see (a)] ; ^^^ _smlxcosx^7 Tgj^es^ia;, [see (6)]. 8 (6) i sin X cos^ a; + f ("cos* x dx, [see (a)]. (c) Jsinxcos'x + tjcossxdx, [see (6)]. 8. (a) _ 1 i^ + § f ^, (see Ex. 4). ° ^ ■' 4 sin* X 4 J sinS x (b) ^secxtanx + logVsecx + tanx. , Cc) lilH^ + f tanx. ^^ 3cos3x ^ (d)iiil^ + |f^, [see(&)]. 4 cos* X 4 J cos^ X l^inx 4r_^_ rsee(c)]. <-*^ 5cos6x^6J cos*x' •• ^ '■' '"' ' 3 sin3 X ' (3) _1^ + K^, [see(/)]. Art. 48 4. (a) I tan3 x + tan x + c. (c) - i cotf x - | cot»x - cot*. (6) - i cots X- cot X. (d) tans| + 3tan|+c. 5. (a) Uanxsecx + ilogtanf J^-|)• f ^ ' /t x\ 1 (b) i:tanxsec3x+ | jtanxsecx + logtanl - + -j j-- ^^^ _ cotxcosecx _^^^Pgt^„x, 2 / a;\ ((^) _ J cot X cosecs X - f f cot X coseo X - log tan - \ • (e) ftanf xsecf x + |logtanfj + |y Art. 49 2. (a) i tanS x - tan x + x. ((Q - ^ cot^ x - log sin x. (b) |tan*x- Uan^x + logsecx. (e) - i cot^x + ootx + x. (c) I tanS .( - i tan3 x + tan x - x. (/) - J cot* x + | cot2 x + log sin x. ANSWERS 276 2. Art. 50 5 11 li 1 U. I swJ X — ^ sin'T x + ^j sin ^ a;. 3. — ^ cos' x + y\ cos^^ x. . 2 - 2vcosa; (1 — |cos2a;+ |cos*a;). 5. |sm' a;(l — Isin^a; + ^sin*a;). J tan^ x-\-\ tan* a; + c. 7. — ^ cot' x — \ cot^ x + c. Art. 51 cosz / sin^g _ sin^a _ sina N , ^ 2^3 12 8/16' 3. tana; — 2cotx — Icot'a;. 4C0S X /-o n \ O it . (3 — cos^'x) 2smx 2 6. - °°^^ -cos a:- 8 log tan-. 2sin2a; ^ 2 Art. 52 I tan^ a; + 2 log tan x — \ cot^ x. c. cosec^ x 3. ^i tan' J' a; + f tan^ x. 4. — 4 cot« x — \ cof X. l/3x_sin2a;+'"^'=^ t\2 , -^v 4V 2 8 y 4. TV(5a;-4sin2a; + Jsm32a; + f sin 4a;). 5. yij (5 a; + 4 sin 2 a; — I sin^ 2 a; + -| sin 4 a;). 5. - ii^fiiiLJi + f cosec' a; - i cosecs a;. 6. ^tan9a; + ^-tan'x. 3 7. 4 sec* a; (^j sec'' a; — y^j sec a; ,+ 1) . Art. 53 3. f x + Jsin2a;+ j'jsin4a;. sin^ 2 X ,x__ sin4x " 48 16 64 ' 7. T|y(3x — sin4x + - (3 tan :5 _ 2) 1. _?-tan-i. ^ V5 1 log V5 2 tan x + 3-v/5 2^5 '°2tanx + 3+V5 itan-i/itan|y Art. 55 4. ^tan-i[2tan-y tan X + 3 5. J log 6. I log tan X — 3 2 tan X + 1 tan X + 2 1. Je^'Csinx — cosx). 2. |e»^(sinx + cosx). 3. tV e^'^CS sin 3 X + 2 cos 3 x) . INTEGRAL CALC. — 19 Art. 56 4. tV «"*"(2 sin 2 X - 3 cos 2 x) 5. — ^ e-«(sinx + cosx). 2 sin 2 X + cos 2 6. ie'(l + ' ^)- 276 ANSWEBS Art. 57 ^ cos 8a: cos 2 x 3 sin 11 a: . sing 16 4 ' ■ 22 2 ' • 11 ■ o 4. A sin iJ- x+ A sin | x. „ sin 1 1 X , sm 3 X ^^ 3 ' 10 j 22 6 ' 5. — J cos X — cos J x. 6. — ^ sin X + I sin | x. CHAPTER VIII Art. 59 4. 240. 5. 80. 6. The double infinity of straight lines y = cix+ Cz, in which Ci, Ca are arbitrary constants. 7. 3j/ = = 2x(x2- !)• 8. Art. 60 9. n(p - a). 3. 24-^ 6. -vv-«'-" 7. 6 68. 4. 58^. 6. Art. 62 8. 32 a'. 4. (^2 4c -62) 6. «^<'. 90 5. fa' tana. r-fC^-Das. CHAPTER IX Art. 65 e 2. cr = eK 3. r» = csinne; }' = csin9; r= c(l — cosS). Art. 66 3. i a2((33 - .a»). 4 8a2 3 Art. 69 5. 1 6. 3|!. 3. Trab. Art. 70 4. 5,r2as. 3. 2 TrV^fe. 5. iwa%. 7. {T-i)a^h. 4. jm. 6. ^ cubic feet. 8. TrVpqh\ ANSWERS 277 Art. 71 4a 2a ' = y/ax, + a:i^ + glog ^ + ^" + '"i. Va s = 2.29558 a. 5. s = 4a( cos -2 — cos J ) ; length of a complete arch = 8 a. 6. s = 6 a. Art. 72 3. 8a. 4. g reaVrrg? - gi VnM? + log ^-±-^1 +-gg"|. 2L , Si + Vl + fliJ U2(a+Va2 + »-i2)J a 7. 2a/V5-2-V31og ^ + >^ I. l V2(2 + V3) ) 8. s = atan Jsec J + alogtan (-i + -); s = 2a(sec-+ logtan IttV 22 \44/ \4 / Art. 73 1. s = r(p. 2. s = 4a( 1 — COS-)' 3. (1) s = 4a(l — COS0); (2)s = 4asln0. 4. (1) s =ptan0sec0 +plogtan (£ + —); (2) s=j9tan/,^ + ^^seo|.^+lWplogtan^|+^^-i)V2-;)log2_pv^. 5. 9s = 4a(sec'0 — 1). 8. s = clogsec0. 6. s = alogtan(| + l). ^ . = i«sin^^. 7. s = a(e=* - 1). 278, ANSyVERS Art. 74 2. 2Tb(b+ "^ cos-'-V or 3. f7raJ(a + x)^- \ Va^ - 62 aj I IN . , 4. 27ra2(i_i]. \ e / 5. J/Tra^. in which e Is the eccentricity. 8- t (t — 2) a^. 7. V Ta^. Art. 75 3. 2(ff - 2)a2. 4. - Art. 76 2 7ra 'Jab 3. 26|. 6. .7854 a. 10. --1", or 32.704°. 4. 5. (a) 9? ; (6) (c) -iI'^^ ^ '^ 60 62 12J; 7. 8. 9. .6366 a. .6366. 1.273 a. Art. 77 11. 2 im2. 2. a 2' 3. 4. Page 164 5. 6. fa. 1. fTraS. ,e = I Tra^fi ; surl centricity. a'^i„_2x + a + 2 Vax + x^ i 7r62 1 4- e 2. Volume = % ira^h ; surface = 2 wa'^ -\ log , in which e is the .... e 1 — e eccentricity. 3. IT Ux + ~\Vax + x^ -^\og 4. 4 ir^a'. 8. irWa'^ + 6^. 6. 4 Tr^aft. 7. |a62cota. 8. f a^A. 9. Volume = J/ a^ ; surface = 8 a^. 10. 3 ^^s. n. j, ^^s. , „ IT- 1 o q / 2 sin o , sin a cos'2 a \ 12. Volume = 2 wa^ a cos a | : I 3 3 )' surface = 4 ira'2(sin o — a cos a). 13. Volume = ir^a^ ; sui-face = Y- ■n-a'^. 14. Surface = ^-^ tto,^. 15. Volume = ■^ns/' — - - V surface = 8 7ra2(ir - f ). 16. !l^*-!(i0-3 7r). 17. J5^(3o2 + 4o6 + 8 62)/i. 6 ANSWERS 279 18. 2aV3. 19. g = V6-'' + y' + b log ^^'^ + V^ - b ^ ^^^^j.^ b=-^- y log a 20. 8&(a + 6) , 21. (^) 2(8-V27) (6) 2(V27 - l)p. • a 3V3 22. logV3. 23. ^. 24. ^-^- 25. ^[W + 4)^-8]. 2 2 o CHAPTER X Art. 79 5. The density at a point three fourths of the distance from the vertex to the base, namely, \kh. 6. x = \h. 7. x=|ft; S = fft. 8. x = y = la. 9. !c = 5. 11. x='^, y = 0. 12. Mass =i I te^, if density = k distance. Mean density = .4244 max. density. Center of mass is aiy = 0, x = j\ ira = .589 a. ,„ _ „ _ 2c(sina 1- - 4a - 4 6 13. 2/ = 0, a; = - — 14. x=— , y = -— 15. (a)x = |A, F = 0; (6) K = I A, ^ = I A, in which k is the ordinate corresponding to * = ft. 17. ic = j? = |ff.-- 18. x = 2/ = ia. 19. x = |a, ?^ = 0. TT 20. X = — I a, ^ = 0. 24. At a point distant f a from the base. 21. x = |a;, ^ = 0. 25. x = — fa, y = 0. Art. 80 (In the answers M denotes mass) 4. If a is the radius, I=\ Ma?, k = — • V2 j_ a6(a^ + 6^) g. ^j.^^. 7. a6f. 12 V6 20 8. {a) I=\Mh'^; (J))I=\Ma^; (c) 7 = J M (a^ + 52) . 9. I=M—- 10. /=^Jf(62 + c2). 11. 7=|ilfa2. 12. l=\Ma\ o 13. / = i^ a6 = I ilfa^ since M=lka' by Ex. 12, Art. 79. 280 ANSWERS CHAPTER XI Art. 82 2 52.4-9 2.4.6. 13 2 02. 4 11 2. 4. 6 16 5. 3 j;f (1 _ , a;2 _ .jl^3;4 _ ^Z^ ^ + ...) + J. 8. 2VsEr.(l+l^ + l^^+...)+c. 9. logx + K + -^ + — ^^ — +... + C. 1. 22 1.2- 32 mx , m'^r? , m^x? ^ 1 1-22 1-2.32 IT'S /V'O »y"7 Art. 83 2. sm-ix = x + l.^ + 2.g-g^+- + l-3-(2''-l) -^!2^+, 2 3 2 4 5 2-4 -2n 2b + 1 5- log(l + x)=f-f + ^-^V..-. 12 3 4 log(l- x) = 1 X2 2 ' X8 3 X* 4 ■■■■ . log(x + = X - 1 2 ' x3 1 - 3 x5 1 - 3 - 3 2-4 5 2.4. 6 7 6. Vl + X2) = CHAPTER XIII Art. 97 4- y-X^+'^ \dx 3 '■"S.*" = 2x|^. dx 5. (1-X2)( ,dx} 2 + 1 = :0. ,.§...,= --0. ^+X f = c. Art. 98 3. tanj/ = c(l 2. yVl —X' VI -, - e')8. ANSWERS 281 Art. 99 x3 2. xj/2 = c2 (k + 2 2/). 3. y = oe^. Art. 100 1. a''x-^-xy^ -x^z^c. 2. x^ ~ 6x^y -6xy^ + y' = c. 3. x^'^ + ix^y — i xy^ + y^ - xe)/ + e^^'y + x^ = c. Art. 101 2. 2a\ogx+ alogy — y = c. 3. x^ — y^ — 1 = ca;. Art. 102 2. y = (x + c)e-«. 4. 2/(x2 + l)2 = tan-ix + c. 1 S. y = x2(l + ce"'). 5. X^ = ax + c. Art. 103 2. 7 2^~3- _ gj.| _ 3 J.3. 3, yY _ c(i _ a;2)i _ 1 -^' . 4.~-«0T/5(x + 1)2 = 10 x6 + 24 x5 + 15x* + c. Art. 104 2. 343(2/ + c)s = 27 ax'. 3. (?/ -c){y + x^- c) {xy + cy + 1) = 0. Art. 105 2. log ( p — x) = 1- c, with the given relation. 3. x = log p^ 4. 6 p + c. Art. 106 1. 2/=c— alog(p — 1), x=c+alog^^- 2. y—c= Vx— x^— tan-i-v/^^- p— 1 1 X Art. 107 3. 2/ = cx + sin-ic. 4. 2/ = cx+— . 5. y2 = cx2 + l+c Art. 108 3. The catenary 2/ = - (e" + e ») or ^ = cosh-i -. 2 a a 4. The envelope of the family of lines y = ex -\ — ^^ , namely the parabola (x-2/)2-2a(x + 2/)+a2 = 0. °~^ 282 ANSWERS 5. The envelope of the lines y=cx+aVl+c^, namely the circle x^+y^=a^. 6. The circles x'' + y'^ = 2 ex. 7. The circles x^ + y^ = k^. 8. The circles which pass through the origin and have their centers on the j/-axis. 9. The rectangular hyperbolas x^ — y^ = d' whose axes coincide in direction with the asymptotes of the former system. 10. x'' + y^ = 2a^logx + c. 18. The system of circles r = c' sin 9 which pass through the origin and touch the initial line. 13. r" cos ne = o". 14. The confocal and coaxial parabolas r = 1 — cos 15. The system of curves j'" = c"sinK9; r^ = c''sin2 6, a series of lemnis- cates having their axis inclined at an angle of 45° to that of the given system. Art. 109 2. x = -^sinnt + At + B. *• y = ix'^(il- \x). 3. x = \gfl + At + B. 5- 2' = a;e'-3e' + cia;2 + c2a; + C3. 6. y = ci + c^x + djfl + ■■■ + c„x"-i + ,^^ |m + n Art. 110 3. ax = log {y + \/y^ + ci) + C2, or y = ci'e'" + C2'e-«». Art. Ill 2. e-'^y = cxx -{■ d. 3. 15 2/ = 8 (x + ci)^ + Cax + cs. Art. 112 1. 2/ = ci log K + C2. 2. y=cisinaa; + C2Cosaa; + C8a; + C4. 3. 2/2 = a;2 + cix + 02. ANSWERS 283 Art. 114 3. x = cie^' + c-ie-*'. i. y = Cie-^ + e*(c2 cos Vs x + Cs s'm VS x). 5. y = cie-' + c^e ^ + cse^. 6. y = cie-^ + cae** + cgt^. 7. y = cie'" + C2e-<" + casin{ax + a). Art. 115 2. y = ci + e-^(c2 + csS). 3. y = e-^^^Oi + c^x + csx^) + C4e*». Art. 116 Z. y = cia;~' + CiX-^. i- y = x\ci + cz log x). 6. 2/ = x2[ci + C2loga; + C3(logx)2]. Page 227 2 a2 = 2a;M — + a;2. 12. !/-»+i=ce("D»i'"^+2sina:+^-. ' (fo; ra— 1 3. lsxi.xis,ay = lc\ 13. — = x2^-l + ce*^ 12/ 2/ 5. X!/(x -y)=c. ^ 6. ax2 + 6xj/ + C2/2 + gra; + ey = fe. 18. 2 y = cx^ + -• 1 7. x = c2/e^. 16. y2 = 2cx + c2. e» 17. e* = ce"^ + c'. 2' 18. (cix + C2)2 + a = Ci^s. X 1 9. y = - i-cx". 19. y = ci + C2X + cse" + CiB- 1 — a a 10. j, = tanx-l + ce-f'". 20. y = c^ - sm-U^e-. 11. 2^ = ax + ex Vl - xK 21. ?/ = Cie-"' + Ca + | e"^. V3 , , „,■„ V3, r = cie-"^ + e2 ( Cacos — x + Cssin— -X j. 23. 2/ = csin (nx + a)(ox+ 6). 24. 2/ = c^(ci + cax) sin x + e'(cs + C4X) cos x. 25. !/ = Ci + C2X + e»(c3 + C4X). 26. y = x(ci cos log X + C2 sin log x) + C3X-1. 27. 2/ = (ci + C2 log x) sin log x + (ca + C4 log x) cos log x. 28. ^72,=|^?|!-^) + ciX + C2. 29. .B72/=-76-^+0ix3 + C2iB2 + c8X + C4. INDEX [The numbers refer to pages.] Algebraic transformations, 103, 105, 107, 108, 113. Amsler, 188. Ainsler's planimeter, 189. Angles, use of multiple, 114. Anti-derivative (anti-differential), 1, 5, 7, 11, 12, 14, 21, 25. Applications to mechanics, 167. Approximate integration, 177-189. rules for, 181-188, 2;«. Areas, change of variable, 141, 142. derivation of integration formulse for, 9, 27. oblique axes, 140. polar coordinates, double integra- tion, 139. polar coordinates, single integra- tion, 135. precautions in finding, 63. rectangular coordinates, 58. rectangular coordinates, double in- tegration, 126. rules for approximate determination of, 181-188, 2.3.5-237. surfaces of revolution, 152-156. surfaces z=f(x, y), 156-160. Auxiliary equation, 223, 224. Bernouilli, James and John, 2. Bertrand, 237. Boussinesq, 186. Cajori, 2. Cardioid, area, 138. center of mass, 173. intrinsic equation, 151. length, 149. orthogonal trajectories, 216. surface of revolation, 156. Carpenter, 189. Catenary, area, 76. intrinsic equation, 150. length, 147. surface of revolution , 156. volume of revolution, 77. Center of mass, 168. Change of variable, 41. Circle, area, 61, 139, 140, 141. evolute of, 166. intrinsic equation, 151. length, 146, 149. orthogonal trajectories, 217. Cissoid, center of mass, 173. length, 149. volume of revolution, 77. Clairaut, 213. equation of, 213. Complementary function, 222. Cone, center of mass, 172. moment of inertia, 175. surface, 164. volume, 71, 73, 143. volume of frustum, 77. Conoid, volume, 143. Constant of integration, 22. geometrical meaning of, 23. two kinds, 25. Convergent, 178. Cotes, 236. Curves, areas of, oblique axes, 140. areas of, polar coordinates, 135-140. areas of, rectangular coordinates, 58. derived, 29. derivation of equations of, 25, 26, 7.n, 134, 214r-217. integral, 33, 190-200, 240-245. intrinsic equations of, 149. quadrature of, 58. Cycloid, area, 141. intrinsic equation, 151. 285 286 INDEX Cycloid, length, 147. note on length, 144. volume of revolution, 142. volumes and surfaces of revolution, 165. Cylinder, 143. moment of inertia, 176. Density, 167. Derived curves, 29. Derivation of equations of curves, 25, 26, 75, 1.S4, 214-217. Derivation of fundamental formulse, 48. Differential equations, see Equation. Differential, integration of total, 52. Differentiation under integration sign, 190. Durand, 186, 187, 190, 237. Equations, homogeneous in x, y, 205. linear, constant coefficients, 223. linear, homogeneous, 225. linear of first order, 208. linear of »th order, 222. linear, properties of, 222. of order higher than first, 218-228. reducible to linear form, 209. resolvable into component equations, 210. solutions, general, 202. solutions, particular, 202. solvable for x, 212. solvable for y, 211. variables easily separable, 205. Evolute, of circle, length, 166. of parabola, length, 165. Expansion of functions in series, 179. Exponential functions, 117. Figures of curves, 246-248. Fisher, Irving, 30. Folium of Descartes, area, 138. ForinuliB, areas, 9, 27, 137, 139, 141. lengths, 145, 148. of approximate integration, 183, 185, 186, 187, 235-237. of integration, fundamental, 37, 47, 48. of integration, table of, 249-262. of integration, universal, 39, 44. of reduction, 46, 93, 94, 95, 101, 102, 106, 110, 231-234. surfaces, 154, 155, 15T, 158. volumes, 70, 130, 143. Fourier, 9. Fractions, rational, 78-83, 229-231. Functions, irrational, 84r-y9. trigonometric and exponential, 100- 118. Geometrical applications, 58-77, 126-166, 214-217. meaning of constant of integration, 23. principle, 7. representation of an integral, 14. Graphical representation of a definite integral, 73. Gray, 189. Gregory, 180. Hele Shaw, 189. Henrici's report on planimeter, 189. Hermann, 188. Hyperbola, area, 68. orthogonal trajectories, 217. related volumes, 77. Hyperbolic spiral, area, 138. length, 149. Hypocycloid, center of mass, 172, 173. intrinsic equation, 152. length, 147. surface, 156. volume of revolution, 76. Integrable form, 37. Integral, complete, 203. Integral curves, 33, 190-200. applications, 195-198. applications to mechanics, 195, 240- 242 applications to engineering and elec- tricity, 242-244. determination of, 198. relations, analytical, 192. relations, geometrical, 194. relations, mechanical, 195. Integral, definite, 8. definite, evaluation by measuring areas, 181. definite, geometrical representation, 14. definite, graphical representation, 73. definite, limits, 9. definite, precautions in finding, 67. definite, properties, 15. definite, relation to indefinite, 24. INDEX 287 Integral, elliptic, 147, 180. general, 23. indefinite, 22. indefinite, directions for finding, 54. multiple, 120. name, 1, 21. particular, 23, 204, 222. Integrals, derivation of, 48. fundamental, 36-38, 47. table'of, 249-2B2. Integraph, 11)8, 200. theory of, 244. Integrating factors, 207. Integration, aided by changing variable, 41, 141. approximate, 177-189, 235-237. by parts, 44, 4H, 100, 231, 233. constants, 22, 202. definition, 1, 18, 21. derivation of reduction formulse,231, 233. fundamental formulae, 37, 47. fundamental rules and methods, 36. in series, 177-179. mechanical, 188, 189, 200. of a total differential, 52. precautions, 63, 67. sign, 2, 21. signs in successive integration, 125. successive, one variable, 119-123. successive, two variables, 123-125. universal formulse, 39, 40, 44. uses of, 1. Intrinsic equation of a curve, 149. Irrational functions, 84-99. Jevons, 30. Lamb, 237. Laurent, 180. Legendre, 180. Leibniz, 2, 59, 144, 180. Lemniscate, area, 1.37. Lengths of curves, pnlar coordinates, 147. rectangular coordinates, 144. Limits of a definite integral, 9. Logarithmic curve, area, 76. length, 165. Logarithmic spiral, area, 137. length, 149. Markoff, 186. Mass, 167. Mass, center of, 168, 169. Mean value, 17, 160-164. definition, 103. Mechanical integration, 188-189, 200. Multiple angles used, 114. integral, 120. Neil, 144. Newton, 2, 59, 73, 144, 180, 236. Oliver, 4. Orthogonal trajectories, 214-217. Parabola, area, 5, 14, 59, 68, 76, 126, 138. center of mass, 173. derivation of equation, 26. intrinsic equation, 151, 152. length, 146, 149. length of evolute, 165. orthogonal trajectories, 216, 217. semicubical, area, 68. semicubical, intrinsic equation, 152 semicubical, length, 144, 147. surfaces of revolution, 155, 164. volume of revolution, 71, 77. Parabolic rule, 184, 186, 236. Paraboloid, center of mass, 173. volume, 131, 143. Pascal, 59. Planimeter described, 188, 189. theory of, 237. Pyramid, volume, 143. Quadrature, 58, 73, Range, 161, 163. Rational fractions, 78-83. decomposition of, 229-231. Reciprocal substitution, 84. Reduction formulfe, 46, 93-95, 101, 102, 106, 110, 231-233. Roberval, 58. Signs of integration, 2, 21, 125. Simpson, Thomas, 184. one-third rule, 182, 184, 186. three-eighths rule, 236. Slope of a curve, 25. Solids of revolution, surfaces, 152-156. volumes, 69. Solutions, general, 202. Sphere, surface, 155, 158. volume, 131, 132, 133, 143, 164. 288 INDEX Spheroid, surface, 155. volame, 72. Spiral of Archimedes, area, 138. length, 149. Substitution, 41, 54. reciprocal, 84. trigonometric, 85. Summation of infinitesimals, 1, 2, 234, 235. Surfaces of revolution, areas, 152-156. volumes, 69. f[x, y, z) = 0, areas, 156-160. /(x, y, z) = 0, volumes, 126-133, 128. Table of integrals, 249-262. Torus, surface, 164. volume, 143. Total differential, integration of, 52. Tractrix, intrinsic equation, 152. Trajectories, orthogonal, 214-217. Transformation, algebraic, 103, 105, 107, 108, 113. Trapezoidal rule, 182, 186, 2.36. Trigonometric functions, integration of, 100-118. substitutions, 85. ^ Undetermined coefficients, use of, 93, 109. Universal formulae of integration, 39, 40, 44. Uses of integral calculus, 1. Volumes, of revolution, 69. polar coordinates, 131. rectangular coordinates in general, 128. Wallis, 59, 144. Weddle, 236. Williamson, 180, 235. Witch, area, 64. volume, 164. Wren, 144. Typography by J. S. Gushing & Co., Norwood, Mass., U. S. A. BY THE SAME AUTHOR DIFFERENTIAL EQUATIONS AN INTRODUCTORY COURSE IN DIFFERENTIAL EQUATIONS FOR STUDENTS IN CLASSICAL AND ENGINEERING COLLEGES 234 pages. $ 1.90 ' The aim of this worli is to give a brief exposition of some of the devices employed in solving diiiferential equations. The boolc presupposes only a knowl- edge of the fundamental formulae of integratiou, and may be described as a chapter supplementary to the elementary works on the integral calculns.' — EsAructfrom Pi-eface. In use as a text-book in Johns Hopkins University, Baltimore, Md.; Van- derbilt University, Nashville, Tenn. ; University of Missouri, Columbia, Mo.; Purdue University, La Fayette, Ind.; Wesleyan University, iMiddletown, Conn.; University of Toronto, Canada; Corijell University, Ithaca, N. Y. ; University of Denver, Denver, Colo. ; Armour Inst., Chicago, 111. ; and other leading institutions. 'We commend the book as providing an excellent introductory course in Differential- Equations.' — American Mathematical Monthly, June, 1897. 'Tlie book seems to be an excellent practical introduction to differential equations, containing a well-proportioned and suitable treatment of most of the topics which the student needs in his tirst course in the subject, and of these only, a good variety of exercises, and enough historical and bibliographical notes to suggest further reading.' — Bulletin of the American Mathematical Society, March, 1898. ' The work, which we have read with considerable interest, assumes in the reader little more than a knowledge of the fundamental formulae of integration, and brings in many practical applications well adapted for the class of students for whom it is intended. The rigorous proofs of many of the theorems are relegated to the appendix to be read when some familiarity with the subject has been acquired. An interesting feature are the numerous liistorical and biographi- cal notes scattered throughout the text, and there are full indexes of names and of subjects which add to its utility.' — The London, Edinburgh, and Dublin Philosophical Magazine, October, 1897. ' Mr. Murray's book is adapted to provide for students that knowledge of the subject of differential equations which they are likely to want in applications of mathematics to physics, and in the general courses in arts and science in "clas- sical " colleges. The author is chiefly occupied with giving expositions of the devices usually employed in the solution of the simple differential equations which such students meet with, and he will be found a safe guide in these mat- ters. He follows the plan, which most recommends itself to teachers, of omitting theoretical considerations, or postponing them until the student has had practice in carrying out the processes with which he must be acquainted before the theory can be understood.' — Nature, 10 Feb., 1898. ' The subject is explained in a simple and intelligible way ; and a very large number of references and interesting historical notes are given, which cannot fail to excite the reader's ambition to learn more of the subject. Another feature is that in some cases a theorem is only stated, while the proof is postponed to a note at the end of the book. This plan, if not employed too often, seems a good one ; and in the case of difficult theorems it would be well if it were more com- monly adopted.' — The Educational Times, 1 Nov., 1897. LONGMANS, GREEN & CO., Publishers 91 & 93 Fifth Avenue, New Turk Higher Mathematics Crockett's Plane and Spherical Trigonometry With Tables . $1 .25 The Same Without Tables . . . . 1 .00 Logarithmic and Trigonometric Tables-(separate) . . 1.00 A clear analytic treatment of the elements of Plane and Spherical Trigonometry and their practical applications to Surveying, Geodesy, and Astronomy, with convenient and accurate "five place " tables for the use of the student, engineer, and surveyor. Designed for High Schools, Colleges, and Technical Institutions. Raymond's Plane Surveying Cloth, 8vo, 485 pages. With Tables and Illustrations . $3.00 A modern text-book for the study and practice of Land, Topograph, ical, Hydrographical, and Mine Surveying. 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