B ^7o '^5 i!!;i!jl;i:ii:';f;;;Oih't'''';;f m , .ji;|,|i,!liiiiii;:'i|;!:.|;;,;:;i;^i:;v, ISilliiiiiiiiiii^illii?!'' ti;' ;;■ m^^. 'lli' liia;:.^i:iii!l:S|f I: iiii , , I 'Ii I Nli I'll "'!'•' ' I llil^ IS iill ill: IM iiiiiiiiiifi; From the Library of Professor Arthur Hall donated by- Major Douglas Merkle ?7075 CORNELL UNIVERSITY LIBRARY GIFT OF Professor Joe Bail Cornell University Library QA 303.081 Differential and integral calculus, with 3 1924 015 990 108 The original of tliis book is in tlie Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924015990108 DIFFERENTIAL AND INTEGRAL CALCULUS WITH EXAMPLES AND APPLICATIONS BY GEORGE A. OSBORNE, S.B. WALKEK PROFESSOR OP MATHEMATICS IN THE MASSACHUSETTS INSTITnTE OP TECHNOLOOT JAN 1-. 1977 .1 '~Rt:VISED"EDiTION" BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHIiRS 1906 Copyright, 1891 and 1906, By GEOKGE a. OSBOKNE PREFACE In the original work, the author endeavored to prepare a text- book on the Calculus, based on the method of limits, that should be within the capacity of students of average mathematical ability and yet contain all that is essential to a working knowledge of the subject. In the revision of the book the same object has been kept in view. Most of the text has been rewritten, the demonstrations have been carefully revised, and, for the most part, new examples have been substituted for the old. There has been some rearrangement of subjects in a more natural order. In the Differential Calculus, illustrations of the " derivative " have been introduced in Chapter II., and applications of differentia- tion will be found, also, among the examples in the chapter imme- diately following. Chapter VII., on Series, is entirely new. In the Integral Calculus, immediately after the integration of standard forms. Chapter XXI. has been added, containing simple applications of integration. In both the Differential and Integral Calculus, examples illustrat- ing applications to Mechanics and Physics will be found, especially in Chapter X. of the Differential Calculus, on Maxima and Minima, and in Chapter XXXII. of the Integral Calculus. The latter chap- ter has been prepared by my colleague, Assistant Professor N. E. George, Jr. The author also acknowledges his special obligation to his col- leagues, Professor H. W. Tyler and Professor ¥. S. Woods, for important suggestions and criticisms. CONTENTS DIFFERENTIAL CALCULUS CHAPTER I Functions akt8. pages 1. Variables and Constants . . l 2-7, 9. Definition and Classification of Functions .... 1-5 8. Notation of Functions. Examples 5-7 CHAPTER II Limit. Increment. Dekivative 10. Definition of Limit 11. Notation of Limit 12. Special Limits (arcs and chords, the base e) 13-15. Increment. Derivative. Expression for Derivative . 16. Illustration of Derivative. Examples . 17-21. Three Meanings of Derivative . ... 22. Continuous Functions. Discontinuous Functions. Examples CHAPTER III Differentiation 24-32. Algebraic Functions. Examples .... 33-38. Logarithmic and Exponential Functions. Examples 39-46. Trigonometric Functions. Examples . 47-55. Inverse Trigonometric Functions. Examples 56. Relations between Certain Derivatives. Examples 8 . 8- -10 . 11 12 13- -15 16- -21 22- -25 26-39 39-45 45-51 51-57 57-60 CHAPTER IV Successive Differentiation 57, 58. Definition and Notation 61 59. The nth Derivative. Examples . 63-65 60. Leibnitz's Theorem. Examples 65-67 VI CONTENTS CHAPTER V Differentials. Infinitesimals ARTS. 61-63. Definitions of Differential 64. Formulae for Differentials. 65. Infinitesimals Examples . PAGES 68-70 71-73 73,74 CHAPTER VI Ijiplicit FnNOTioNS Differentiation of Implicit Functions. Examples 75-77 CHAPTER VII Series. Power Series 67, 68. Convergent and Divergent Series. Positive and Negative Terms. Absolute and Conditional Convergence . . 78, 79 69-71. Tests for Convergenpy. Examples .... 79-85 72, 73. Power Series. Convergence of Power Series. Examples . 85-87 CHAPTER VIII Expansion of Functions 74-78. Maclaurin's Theorem. Examples 7'.). Huyghens's Approximate Length of Arc 80,81. Computation by Series, by Logarithms 82. Computation of tt . 83-87. Taylor's Theorem. Examples 89. RoUe's Theorem 90-93. Mean Value Theorem 94. Remainder 88-93 93 94-96 96,97 97-100 101 101-104 105 CHAPTER IX Indeterminate Forms 95. Value of Fraction as Limit . ... 96,97. Evaluation of -. Examples 98-100. Evaluation of g, 0-x, 00-00, 0», 1", 00°. Examples 106 106-110 110-113 CONTENTS vii CHAPTER X Maxima and Minima op Functions of One Independent Vakiablb ARTS. PAGES 101. Defimtion of Maximum and Minimum Values ... 114 102-104. Conditions for Maxima and Minima. Examples . . . 114-119 105. When ^ = ®. Examples ....... 119-121 106. Maxima and Minima by Taylor's Theorem. Problems . 121-129 CHAPTER XI Partial Difperentiation 107. Functions of Two or More Independent Variables . . 130, 131 108. Partial Differentiation. Examples 131 132 109. Geometrical Illustration I33 110. Equation of Tangent Planes. Angle with Coordinate Planes. Examples 133-136 111, 112. Partial Derivatives of Higher Orders. Order of Differentia- tion. Examples 136-139 113. Total Derivative. Total Differential. Examples . . 140-144 114-116. Differentiation of Implicit Functions. Taylor's Theorem. Examples 144-147 CHAPTER XII Change op the Variables in Derivatives 117. Change Independent Variable xtoy 148, 149 118. Change Dependent Variable I49 119. Change Independent Variable x to z. Examples . . 150-152 120, 121. Transformation of Partial Derivatives from Rectangular to Polar Coordinates 152-154 CHAPTER XIII Maxima and Minima of Functions of Two or More Variables 122, 123. Defimtion. Conditions for Maxima and Minima . . 155, 156 124. Functions of Three Independent Variables .... 166-161 CHAPTER XIV Curves for Reference 125-127. Cissoid. Witch. Folium of Descartes .... 162, 163 Vlll CONTENTS ARTS. 128-130. Catenary. Parabola referred to Tangents. Cubical Pa- rabola. Semioubical Parabola 131-134. Epicycloid. Hypocycloid. (-Y+(^Y = '^' a'2/^=a'^^*-^'^ 135-145. Polar Coordinates. Circle. Spiral of Archimedes. Hyper- bolic Spiral. Logarithmic Spiral. Parabola. Cardioid. Equilateral Hyperbola. Lemniscate. Eour-leaved Rose. r = asin3- 164, 165 166, 167 167-172 CHAPTER XV Direction of Ccetes. Tangents and Normals 146. Subtangent. Subnormal. Intercepts of Tangent 147. Angle of Intersection of Two Curves. Examples 148. Equations of Tangent and Normal. Examples 149-151. Asymptotes. Examples . ... 152, 153. Direction of Curve. Polar Coordinates. Polar Subtangent and Subnormal .... . . 154. Angle of Intersection, Polar Coordinates. Examples . 155, 156. Derivative of an Arc . 174- -176 176- -179 179- -182 182, 183 183- -186 186- -188 CHAPTER XVI Direction of Curvature. Points of Inflexion 157. Concave Upveards or Downvfards . 158. Point of Inflexion. Examples 189 190-192 CHAPTER XVII Curvature. Radius of Curvature. E volute and Involute 157-161. Curvature, Uniform, Variable 162-164. Circle of Curvature. Radius of Curvature, Rectangular Co- ordinates, Polar Coordinates. Examples 166. Coordinates of Centre of Curvature .... 166, 167. Evolute and Involute 168-170. Properties of Involute and Evolute. Examples . 193, 194 195-200 200 201, 202 202-205 CHAPTER XVIII Order of Contact. Osculating Circle 171, 172. Order of Contact .... . . 173. Osculating Curves 206-208 208, 209 CONTENTS IX 174. Order of Contact at Exceptional Points .... 175. To find the Coordinate of Centre, and Radius, of the Oscu- lating Circle at Any Point of the Curve .... 176. Osculating Circle at Maximum or Minimum Points. Ex- amples . . PAGES 209 209-211 211-213 CHAPTEK XIX Envelopes 177. Series of Curves ... .... 214 178, 179. Definition of Envelope. Envelope is Tangent . . 214, 215 180-182. Equation of Envelope . . . . . 215-217 183. Evolute of a Curve is the Envelope of its Normals. Examples 217-221 INTEGRAL CALCULUS CHAPTER .XX Integration op Standard Forms 184, 185. Definition of Integration. Elementary Principles . . 223-225 186-190. Fundamental Integrals. Derivation of Formulae. Examples 225-240 CHAPTER XXI Simple Applications or Integration. Integration Constant op 191,192. Derivative of Area. Area of Cui-ve. Examples 195. Illustrations. Examples 241-244 244-248 CHAPTER XXII Integration of Rational Fractions 194, 195. Formulse for Integration of Rational Functions. Preliminary Operations ... .... 249 196. Partial Fractions . . _ 250 197. Case I. Examples . ' . . . 250-253 198. Case II. Examples . . 254-256 199. Case III. Examples ... ... 256-259 200. Case IV. Examples 260-262 CONTENrS CHAPTER XXIII Integration of Irkational Functions ARTS. 202. Integration by Rationalization p r 203, 204. Integrals containing ( ax + 6)', (ax + b)'. Examples 206, 207. Integrals containing Vi x* + ax + 6. Examples 208. Integrable Cases PAOB8 263 263-266 266-268 268 CHAPTER XXIV Tkigonometeic Eorms readily Integrable 209-211. Trigonometric Function and its Differential. Examples . 270-272 212,213. Integration of tan" x(?x, cot«xdx, sec" X(ix, cosec"xcto . 273,274 214. Integration of tan™ X sec" X dx, cot*" X cosec" X (Jx. Examples 274-276 215. Integration of sin" X cos" xdx by Multiple Angles. Examples 276-278 CHAPTER XXV Integration by Parts. Reduction FoRMnL.E 216. Integration by Parts. Examples 279-282 217. Integration of C'' sin MX dx, 6°"= cos mx dx. Examples . . 283,284 218-222. Reduction Formulse for Binomial Algebraic Integrals. Deri- vation of Formulae. Examples 284-291 223, 224. Trigonometric Reduction Formulae. Examples . . 291-294 226. 227. 228-232. 283. CHAPTER XXVI Integration by Substitution Integrals of /(x2)xcZx, containing (a + 6x2)9. Examples 295,296 Integrals containing Va^ — x^, Vx" ± a^ by Trigonometric Substitution. Examples . . 296-299 Integration of Trigonometric Forms by Algebraic Substitu- tion. Examples . . . . 299-304 Miscellaneous Substitutions. Examples .... 304, 305 CHAPTER XXVII Integration as a Summation. Definite Integrals 234. Integral the Limit of a Sum 306 235-237. Area of Curve. Definite Integral. Evolution of Definite Integral ,306-309 CONTENTS XI 238,239. Definition of Definite Integral. Constant of Integration. Examples 310-314 240-242. Sign of Definite Integral. Infinite Limits. Infinite Values of/(*) 314-317 243-245. Change of Limits. Definite Integral as a Sum . . 317-319 CHAPTER XXVIII Application or Isttegkation to Plane Cukves. Application to Certain Volumes 246, 247. Areas of Curves, Rectangular Coordinates. Examples 320-324 248. Areas of Curves, Polar Coordinates. Examples . . 325-327 249. Lengths of Curves, Rectangular Coordinates. Examples 327-330 250. Lengths of Curves, Polar Coordinates. Examples . . 330-332 251. Volumes of Revolution. Examples .... 333-335 252, 253. Derivative of Area of Surface of Revolution. Areas of Sur- ■ faces of Revolution. Examples . . . 336-339 254. Volumes by Area of Section. Examples . 340-342 CHAPTER XXIX Successive Integration 255-257. Definite Double Integral. Variable Limits. Triple Integrals. Examples 343-345 CHAPTER XXX . Applications op Double Integration 258-262. Moment of Inertia. Double Integration, Rectangular Co- ordinates. Variable Limits. Plane Area as a Double Integral. Examples . . . 26.3-265. Double Integration, Polar Coordinates. Moment of Inertia, Variable Limits. Examples 266. Volumes and Surfaces of Revolution, Polar Coordinates. Examples . . . . ... 346-350 350-353 353, 354 CHAPTER XXXI Surface, Volume, and Moment of Inertia of Any Solid 267. To find the Area of Any Surface, whose Equation is given between Three Rectangular Coordinates, x, y, z. Ex- amples .......... 355-360 xu CONTENTS ARTS. PAGES 268. To find the Volume of Any Solid bounded by a Surface, whose Equation is given between Three Rectangular Co- ordinates, K, y, z. Examples . . ... 361-363 269. Moment of Inertia of Any Solid. Examples . . . 363, 364 CHAPTER XXXII Centre op Gravity. Pressure of Fluids. EoBCE OF Attraction 270,271. Centre of Gravity. Examples . . ... 365-369 272, 273. Theorems of Pappus. Examples 369, 370 274. Pressure of Liquids. Examples . . . 370-373 275. Centre of Pressure. Examples . . . 373-375 276. Attraction at a Point. Examples . . . 375-377 CHAPTER XXXIII 277. Integrals for Reference . . 378-385 Index 386-388 DIFFEREI^TIAL CALCULUS CHAPTER I FUNCTIONS 1. Variables and Constants. A quantity ■which may assume au unlimited number of values is called a variable. A quantity whose value is unchanged is called a constant. For example, in the equation of the circle X and y are variables, but a is a constant. For as the point whose coordinates are x, y, moves along the curve, the values of x and y are continually changing, while the value of the radius a remains unchanged. Constants are usually denoted by the first letters of the alphabet, ffl, h, c, a, p, y, etc. Variables are usually denoted by the last letters of the alphabet, ^, y, «, , "A; etc. 2. Function. When one variable quantity so depends upon an- other that the value of the latter determines that of the former, the former is said to be a function of the latter. For example, the area of a square is a function of its side ; the volume of a sphere is a function of its radius ; the sine, cosine, and tangent are functions of the angle ; the expressions x% log(a? + l), Vx(x + 1), are functions of x. 1 2 DIFFERENTIAL CALCULUS A quantity may be a function of two or more variables. For example, the area of a rectangle is a function of two adjacent sides; either side of a right triangle is a function of the two other sides ; the volume of a rectangular parallelopiped is a function of its three dimensions. The expressions are functions of x and y. The expressions xy + yz + zx, -y/^y^- log {a? + y-z), are functions of x, y, and z. 3. Dependent and Independent Variables. If ?/ is a function of x, as in the equations y = x', 2/ = tan 4 a;, 2/ = e* + 1, X is called the independent variable, and y the dependent variable. It is evident that when y is a function of x, x may be also regarded as a function of y, and the positions of dependent and independent variables reversed. Thus, from the preceding equations, x=-Vy, a; = ^tan-^2/, a; = log, (?/ - 1). In equations involving more than two variables, as z + x—y = 0, w + wz + zx + y = 0, one must be regarded as the dependent variable, and the others as independent variables. 4. Algebraic and Transcendental Functions. An algebraic function is one that involves only a finite number of the operations of addi- tion, subtraction, multiplication, division, involution and evolution with constant exponents.* All other functions are called transcen- dental functions. Included in this class are exponential, logarithmic, trigonometric or circular, and inverse trigonometric, functions. Note. — The term " hyperbolic functions " is applied to certain forms of exponential functions. See page 00. * A more general definition of Algebraic Function is, a function whose rela,- tion to the variable is expressed by an algebraic equation. FUNCTIONS 3 5. Rational Functions. A polynomial involving only positive integral powers of x, is called an integral function of x ; as, for example, 2 + a;-4a;2 + 3 cc^ A rational fraction is a fraction whose numerator and denominator are integral functions of the variable ; as, for example, 3af + 2a;-l x* + x'~2x' A rational function of x is an algebraic function involving no frac- tional powers of x or of any function of x. The most general form of such a function is the sum of an integral function and a rational fraction ; as, for example, 3a;2-2a; 2 a;2 + a; - 1 + a;3_2a;^-|-l 6. Explicit and Implicit Functions. When one quantity is ex- pressed directly in terms of another, the former is said to be an explicit function of the latter. For example, y is an explicit function of x in the equations y = x^ + 2x, y = Va;' -|- 1. When the relation between y and x is given by an equation con- taining these quantities, but not solved with reference to y, y is said to be an implicit function of x, as in the equations axy + bx + cy + d = 0, y + logy = x. Sometimes, as in the first of these equations, we can solve the equation with reference to y, and thus change the function from implicit to explicit. Thus we find from this equation, bx + d y= — ax + c 7. Single-valued and Many-valued Functions. In the equation y = x^ — 2 X, for every value of x, there is one and only one value of y. Expressing a; in terms of y, we have a;=l± V2/-I-I. 4 DIFFERENTIAL CALCULUS Here each value of y determines two values of x. In the former case, 2/ is a single-valued function of x. In the latter case, a; is a two-valued fun ct ion of y. An m-valued function of a variable a; is a function that has n values corresponding to each value of x. The inverse trigonometric function, tan~^ x, has an unlimited num- ber of values for each value of x. 8. Notation of Functions. The symbols F{x),f(x),(j>(x),\p(x), and the like, are used to denote functions of x. Thus instead of "y is a function of x," we may write y=f(x), or y = {x). A functional symbol occurring more than once in the same prob- lem or discussion is understood to denote the same function or operation, although applied to different quantities. Thus if f{x)=^ + &, (1) then f(jj) = 2/^ -f 5, /(a) = a^ + 5, /(a + 1) = (a + 1)2 + 5 = a^ + 2 a -f 6, /(2) = 2^ + 5 = 9, /(I) = 6. In all these expressions /( ) denotes the same operation as de- fined by (1) ; that is, the operation of squaring the quantity and adding 5 to the result. Functions of two or more variables are expressed by commas be- tween the variables. Thus if f{x, y)=x^-\-3 xy — y^, then f(a, 6) = a^ + 3 a6 - b^ f(b, a)=W-\--dha- a\ /(3, 2) = 3^ + 3.3-2 - 2-' = 23. /(a, 0) = a\ If ^{x,y,z) = a? + yz-y'-\-2, then {^), ■ (1) and if from this relation we express x in terms of y, so that « = 4'(y), (2) then each of the functions {x), Here i//, the cube root function, is the inverse of <^, the cube function. If y = a' = ^{^), then X = log^y = >p(y). Here \j/, the logarithmic function, is the inverse of (*) (3) „ 2 From this we derive x = '— ^ — ^ = il/(y) (4) Here i// as defined by (4) is the inverse of <^ as defined by (3). The notation ^"^ is often employed for the inverse function of <^. Thus, if y = {x), x = ~\y). If y=m, x^f-Hy). The student is already familiar with this notation for the inverse trigonometric functions. If y = sin X, x = sin~^ y. EXAM PLES 1. Given 2ay' — 2xy + y' = a'; change y from an implicit to an explicit function. Ans. yz=x± ^a? — x'. DIFFERENTIAL CALCULUS 2. Given sin (a; — y) = m sin y ; change y from an implicit to an explicit function. Ans. y = tan~^ m 4- cos X 3. Given f(x) = 2x' -3x^ + x + 2; find /(I), /(2), /(L), /(- 1), /(O). Show that /(x + l) -f(x) = 6 a;^, f(x + h) =f(x) + (6 a^ - 6 a; + l)/i + (6 a; - 3)7r + 2 /il 4. Given F (x) = (of - 1)- ; show that i?'(a; + 1) - F(x - 1) = 8 a!^. 5. Given/(a;) = ^^^^i^; &ud f (0), f(x)+f (-x). Show that / (2 a;) -/ ( - 2 a;) = [/ (a;)]^ - [/ (- x)y. 6. If 'l>(0) = e', ^ (« + 6) = <^ (a) (&). Show that the same relation holds for the function if; (6) = cos + V^n^ sin e, giving il/{a + b) = il/{a)il/(b). show that the inverse function is of the same form. 8. If (m, n + 1) + <^ (m + 1, n) = (m + l,n + 1). 12. Given a;, 2/, 2^ / (a;, 2/, z) = z, X, y y, z, X show that f{y + z,z + x,x + y) = 2f{x, y, z). CHAPTER II LIMIT. INCREMENT. DERIVATIVE 10. Limit. When the successive values of a variable x approach nearer and nearer a fixed value a, in such a way that the difference x — a becomes and remains as small as we please, the value a is called the limit of the variable x. The student is supposed to be already somewhat familiar with the meaning of this term, of which the following illustrations may be mentioned. The limit of the value of the recurring decimal .3333 ..., as the number of decimal places is indefinitely increased, is ^. The limit of the sum of the series l+^ + i + -g-H jas the number of terms is indefinitely increased, is 2. The limit of the fraction , as x approaches a, is 3 a^. X — a The circle is the limit of a regular polygon, as the number of sides is indefinitely increased. -The limit of the fraction'—-—, as 6 approaches zero, is 1, provided 6 6 is expressed in circular measure. 11. Notation of Limit. The following notation will be used : "Lim,^" denotes "The limit, as x approaches a, of." For example, Lim^^„^; = 2. XT — ax Lim^^o (2 a^ - lix + h^) =2 a?. 12. Some Special Limits. There are two important limits required in the following chapter. (a) Lim^^ — - — , & being in circular measure. a LIMIT 9 Let the angle A0A' = 2e. and let a be the radius of the arc AC A'. From geometry, ABA' < ^IC'.l' ; that] ! a sin 6 < 2 aO, sin 6 6, ■ n cos 6 ^ > cos e (2) Hence by (1) and (2), 5^ is inter- d mediate in value between 1 and cos 6. As $ approaches zero, cos 6 approaches 1. sin 6 Hence Lim„ 1. The student will do well to compare the corresponding values of 9 and sin 6, taken from the tables, for angles of 5°, 1°, and 10'. Angle e sin 6 1° 10' — = .0872665 36 ^-^ = .0174533 J^^ = . 0029089 .0871557 .0174524 .0029089 (&) Lim^^^ (1 -\ — j_ Before deriving this limit let us compute the value of the expression for increasing values of z. Thus, (1 + 1)^ = 2.25 (1 + 1/ = 2.48832 (1 + -i^yo = 2.59374 (1.01)™ = 2.70481 (1.001)™ = 2.71692 (1.0001)1™ ^ 2.71816 (1.00001)1'"™° = 2.71827 (l.OOOOOl)!™™* = 2.71828 10 DIFFERENTIAL CALCULUS The required limit will be found to agree to five decimals with the last number, 2.71828. By the Binomial Theorem, which may be written -^ f \> i+iy = i+i + ii^+v — iz_v — ^+ z) |2 |3 1 2 When 2 increases, the fractions - , - etc., approach zero, and we have Lim,^(l+^y = l+l+^+^ + ^ + This quantity is usually denoted by e, so that The value of e can be easily calculated to any desired number of decimals by computing the values of the successive terms of this series. Por seven decimal places the calculation is as follows: 1. 2) 1. 3) .5 4) .166666667 5) .041666667 6) .008333333 7) .001388889 8) -000198413 9) .000024802 10) .000002766 11) .000000276 .000000025 e=2.7182818-- This quantity e is the base of the Napierian logarithms. * For a rigorous derivation of this limit, the student is referred to more ex- tensive treatises on the Differential Calculus. DERIVATIVE 11 13. Increments. An increment of a variable quantity is any addi- tion to its value, and is denoted by tlie symbol A written before this quantity. Thus Ax denotes an increment of x, Ay, an increment of y. For example, if we have given y = 3?, and assume x = 10, then if we increase the value of x by 2, the value of y is increased from 100 to 141, that is, by 44. In other words, if we assume the increment of x to be Ax = 2, we shall find the increment of y to be Ay = 44. If an increment is negative, there is a decrease in value. For example, calling a; = 10 as before, \ny = x", if Aa; = — 2, then Ay = — 36. 14. Derivative. With the same equation, and the same initial value of x, a; = 10, let us calculate the values of Ay corresponding to different values of Ax. We thus find results as in the following table. If Aa; = then Ay = and ^y = Ax 3. 69. 23. 2. 44. 22. 1. 21. 21. 0.1 2.01 20.1 0.01 0.2001 20.01 0.001 0.020001 20.001 h 2011 + 71' 20 + /i The third column gives the value of the ratio between the incre- ments of X and of y. It appears from the table that, as Ax diminishes and approaches zero. Ay also diminishes and approaches zero. 12 DIFFERENTIAL CALCULUS The ratio — ^ diminishes, but instead of approaching zero, ap- preaches 20 as its limit. This limit of — ^ is called the derivative of y with respect to x, Ax dy and is denoted by ^. In this case, when a; = 10, the derivative U20. dy - "'^ dx It will be noticed that the value 20 depends partly on the func- tion y = x^, and partly on the initial value 10 assigned to x. Without restricting ourselves to any one initial value, we may ob- tain — ^ from y = re-. dx Increase x by Aa. Then the new value of y will be y' = {x+ Axf ; therefore, Ay = y'-y=(x + Axf — x' = 2a;Aa; + {Ax)\ Dividing by Aw, ^^2x + Ax. Ax The limit of this, when Aa; approaches zero, is 2x. Hence, -^ = 2 a;. da; The derivative of a function may then be defined as the limiting value of the ratio of the increment of the function to the increment of the variable, as the latter increment approaches zero. It is to be noticed that — is not here defined as a fraction, but as dx . a single symbol denoting the limit of the fraction — ^. The student will find as he advances that — has many of the properties of an dx ordinary fraction. The derivative is sometimes called the differential coefficient. 15. General Expression for Derivative. In general, let Increase x by Ax, and we have the new value of y, y' = fix + Ax). DERIVATIVE 13 Ay = y'~y =f{x + ax) -f(x), Ai/_ f(x + Ax)-f(x) - Ax Ax dx Ax Geometrical Illustration. The process of finding the derivative from y = x^, may be illustrated by a square. Let X be the length of the side OP, and y the area of the square on OP. That is, y is the number of squai-e units corresponding to the linear unit of x.' When the side is increased by PP', the area is increased by the space between the squares. That is, Ay=2xAx+(Axy, ^=2a;+Aa;, Ax -^ = LimA^„— ^ =2x. dx Ax 16. From the definition of the derivative we have the following process for obtaining it : (a) Increase x by Aa;, and by substituting x + Ax for x, deter- mine y + Ay, the new value of y. (6) Find Ay by subtracting the initial value of y from the new value. (c) Divide by Ax, giving — • Ax (d) Determine the limit of _ H, as Ax approaches zero. This limit is the derivative dy dx Apply this process to the following examples. EXAMPLES 1. y = 2x>-6x + 5. Increasing x by Aa;, we have y + Ay = 2(x + Axf —6(x + Ax) + 5 ; therefore, A?/ = 2(a; + Aa;)^ — 6(a;4-Aa;)-|-5 — 2a;''-|-6a;— 5 = (6 »^ - 6) Ax + 6 x(Axy + 2(A xf. 14 DIFFERENTIAL CALCULUS Dividing by Ax, -^ = 6 a;2 - 6 + 6 a;Ax + 2 (Aa;V. Aa; J = Lim,^^ = 6a=^-6. ax Ax 2. y = y + Ay = Ay = cc+l x + Ax a; + Aa; + 1 a; + Aa; x Ax a; + Aa; + l a;+l (a; + Aa; + l)(a; + 1) Ay^ 1 Aa; (a; + Aa; + l)(a; + l)' ^ = Lim, „^ dx "^Ax (a; + 1)2 y= Vx. y + Ay= Vx+~Ax, Ay = Va; + Aa; — Vx, Ay _ Va; + Aa; — Va; Aa; Aa; The limit of this takes the indeterminate form -. But bv rationalizing .the numerator, we have Ay _ Ax 1 ^^ Ax(-\/x + Ax+-\/x) Va; + Aa; + Va; dx ''™'^'="Aa; 2Vi 4. y = x*-2x'+3x-i, ^ = 4ar'-4a; + 3. dx 5. y={x-ay, d^ = 3(^_„)2. DERIVATIVE 6. 2/=(< + 2)(3-2«), 7. '=h' 8. II- ™^ n — x 9. 2/ + 2' 10. y-''^'', 15 x + a 11. a = 12. 2/=V^+2; 13. 2/ = a;'^, 14. 2/ = Va= - x', 15. a; = -, 16. Show that the derivative of the area of a circle, with respect to its radius, is its circumference. 17. Show that the derivative of the volume of a sphere, with respect to its radius, is the surface of the sphere. We shall now give some illustrations of the meaning of the deriva- tive. dy dt = -4f-l. dy _ dx 3 x'' dy _ mn dx (n - xf dx 9 dy (y + 2f dy _ dx .1 2a' (x + ay dx _ dt~ t + 1 (t-ir dy _ 1 dx 2Va;+2 dy^ dx 3xi 2 dy _ X dx y/a'-x' dx di~ 1 2ti 16 DIFFERENTIAL CALCULUS 17. Direction of a Plane Curve. This is one of the simplest and most useful interpretations of the derivative. Let P be a point in a curve determined by its equation y — fix), and PT the tangent at P. Let OM=x, MP = y. If we give x the increment A.-B = MN, y will have the in- crement Ay = RQ. Draw PQ. Then tan RPQ (1) ^RQ^Ay ~ PR~ \x Now if Aa; diminish and approach zero, Ay will also approach zero, the point Q will move along the curve towards P, and PQ will approach in direction PT as its limit. Taking the limit of each member of (1), we have Ay^rf^_ Aa; dx at any point of a curve, is the trigono tan TPR = Lim , dx' That is, the derivative metric tangent of the in- clination to OX of the tangent line at that point. This quantity is de- noted by the term slope. The slope of a straight line is the tangent of its inclination to the axis of X. The slope of a plane curve at any point is the slope of its tangent at that point. Thus, -^, at any point of a curve, is the slope of the curve at that point. DERIVATIVE 17 For example, consider the parabola a- = ^py, y = ~- The slope of the curve is *' = — . dx 2p At 0, where x = Q, the slope = 0, the direction being horizontal. At L, where x = 2p, the slope = 1, corresponding to an inclination of 45° to the axis of X. Beyond L the slope increases towards oo, the inclination increasing towards the limit 90°. For all points on the left of OY, x is negative, and hence the slope is negative, the corresponding inclinations to the axis of X being negative. 18. Velocity in Terms of a Variable t denoting Time. A body moves over the distance OP = s in the time t, s being a function oit; it is required to express the velocity at the point P. Let As denote the distance ^ ^ k — '' — ^t PP' traversed in the interval Af. If the velocity were uniform during this interval, it would be equal to — ^ A< As For a variable velocity, — is the average or mean velocity between P and P, and is more nearly equal to the velocity at P the less we make Ai. „ _ . As ds That is, the velocity at P = .Lxm At=o "T; = jI" ds If V denote this velocity, ■" = -^* Thus, — is the rate of increase of s. ' dt Similarly, ^ and -^ are the rates of increase of x and y respectively. 18 DIFFERENTIAL CALCULUS 19. Acceleration. The rate of increase of the velocity v is called acceleration. If we denote this by a, we have by the preceding article, dv « = — -• dt For example, suppose a body moves so that Then the velocity, and the acceleration. dS n ,0 ■y = — = 3 «-, dt dv ' dt' &t. 20. Rates of Increase of Variables. For further illustrations of the derivative, consider the two following problems: Problem 1. A man walks across the street from ^ to B at a uniform rate of 5 feet per second. A lamp at L throws his shadow upon the wall MN. AB is 36 feet, and BL 4 feet. How fast is the shadow moving when he is 16 feet from A? When 26 feet? When 30 feet? Let P and Q be si- multaneous positions of man and shadow. Let AP = x, AQ, = y. Then ^ = ^ = ^_, y = -Al. X PB 36 - a; -^ 36 - When he walks from P to P, the shadow moves from Q to Q'. That is, when Ax=PP', /\y= QQ'. Let At be the interval of time corresponding to Ax and Ay. Ay Then we may write a^^Ax" ■■'■■•• ^ (2) 'At (1) DERIVATIVE 19 If now -we suppose Ait to diminish indefinitely, Ay and Ax will also diminish indefinitely, and we have for the limits of the two members of (2), dx dy di dx dt rate of increase of y rate of increase of x Art. 18. That is, velocity of shadow at any point Q dy velocity of man ~" dx Finding the derivative of (1), we have dy _ 144 dx (36 — xy Hence, velocity of shadow at any point Q = 144 144 (36 — xf See Ex. 8, Art. 16. (velocity of man) (36 - xy 720 (5 feet per second) feet per second (36 - xy = 1.8 feet per second, when a; = 16 ; = 7.2 feet per second, when a; = 26 ; = 20 feet per second, when x = 30. Problem 2. The top of a ladder 20 feet long rests against a wall. The foot of the ladder is moved away from the wall at a uniform rate of 2 feet per second. How fast is the top moving, when the foot is 12 feet from the wall ? When 16 feet from the wall ? Suppose PQ to be one position of the ladder. Let AP=x, AQ = y. Then 2/=V400-a;l (3) 20 DIFFERENTIAL CALCULUS When the foot moves from P to P', the top moves from Q to Q'. That is, if A* = PP', Ay = QQ'. In the same way as in Problem 1, Ay Ay At Ax ~ Aa; At dy dy dt . dx " dx ' dt velocity of top at Q dy velocity of foot dx And from this, that is, From (3), ^ =— =J1=- See Ex. 14, Art. 16. Hence, velocity of top at any point Q = == (velocity of foot) V400— a;^ = 1^=^ feet per second. V400-a;^ The negative sign is explained by noticing from the figure that y decreases when x increases. Hence the rates of increase of x and y have different signs. When X = 12, velocity of top = — 1^ feet per second. When X = 16, velocity of top = — 2| feet per second. Aw From these problems it appears that, while -^ is the ratio between dv the increments of y and x, -r- is the ratio between the rates of increase of these variables. DERIVATIVE 21 21. Increasing and Decreasing Functions. If the derivative of a function of x is positive, the function increases when x increases ; and if the derivative is negative, the function decreases when x increases. For if the derivative -2, which is the ratio between the rates of dx increase of the variables (see conclusion of Art. 20), is positive, it follows that these rates must have the same sign ; that is, y increases when X increases, and decreases when x decreases. But if — is negative, the rates must have different signs ; that is, dx y decreases when x increases, and increases when x decreases. This is also evident geometrically by regarding -^ as the slope of a curve. As we pass from Ato B, y increases as x increases, but from B to C, y decreases as x in- creases. Between A and B the slope -^ is positive ; be- tween B and C, negative. In the former case y is said to be an increasing function ; in the latter case, a decreasing function. For example, consider the function ?/ = a?, from which we find -^ = 3a^. Since ^ is positive for all values of x, the function y = a? is an dx increasing function. If we take y = -.we find -^ = -„■ X' dx x' Here we have a decreasing function with a negative derivative. Another illustration is Ex. 1, Art. 16, CiX When X is numerically less than 1, y is a. decreasing function. When X is numerically greater than 1, y is an increasing function. 22 DIFFERENTIAL CALCULUS 22. Continuous Function. A function, y=f(x), is said to be continuous for a certain value x^, of x, when 2/0 = /(a;o) is a definite quantity, and A^o approaches zero as Axo approaches zero, Aajo being positive or negative. The latter condition is sometimes expressed, "when an infinitely small increment in x produces an infinitely small increment in y." The most common case of discontinuity of the elementary functions (algebraic, exponential, logarithmic, trigonometric and inverse trigo- nometric, functions) is when the function is infinite. For example, consider the function y = ■ which is continuous for all values of x except x= a. When x = a, y= CO, that is, y can be made as great as we please by taking x sufficiently near a. Also when x<,a, y is negative, and when a;>a, y is positive. There is a break in the curve when x = a, and the function is said to be discontinuous for the value x = a. DERIVATIVE 23 The function y = - — is likewise discontinuous when x = a. (x - ay This function being positive for all values of x, the two branches of the curve are above the axis of x. Likewise the functions, tan x, sec x, are discontinuous when x = —- In general, iif{x) =00, when x = a, there is a break in the curve y=f(x) corresponding to a; = a, and both the curve and the function are then discontinuous. i 2* + 2 Another form of discontinuity is seen in the function y =: — , when a; = 0. 2 4-1 Here y approaches two limits, according as x approaches zero through positive or negative values. T 2V2 , Lim,= + „^^=1. Lim,, = — 2'+2_o 2^+1 2'+l We see that when a; = the curve jumps from y=2 to 2/=l, that is from B to A. The function is discontinu- ous for x = 0. It is to be noticed that the Y B A - definition of the derivative C ) X 24 DIFFERENTIAL CALCULUS implies the continuity of the function. For — ^ cannot approach a Ace limit, unless Ay approaches zero when Ax approaches zero. The converse is not true. There are continuous functions which have no derivative, but they are never met with in ordinary practice. EXAMPLES a^ 1. The equation of a curve is 2/ = -5- — a;^ + 2. o (a) Find its inclination to the axis of x, when x = 0, and when x = l. (6) Find the points where the curve is parallel to the axis of X. Ans. a;=0 and x=2. (c) Find the points where the slope is unity. Ans. a; = (l±V2). (d) Find the point where the direc- tion is the same as that at a; = 3. Ans. 0° and 135°. Ans. x = — l. 2. In Problem 1, Art. 20, when will the velocity of the shadow be the same as that of the man ? Ans. When AP =2i ft. When one quarter, and when nine times, that of the man ? Ans. When AP = 12 ft., and 32 ft. 3. A circular metal plate, radius r inches, is expanded by heat, the radius being expanded m inches per second. At what rate is the area expanded ? Ans. 2 in-m sq. in. per sec. 4. At what rate is the volume of a sphere increasing under the conditions of Ex. 3 ? Ans. i Trj-^m cu. in. per sec. volume is increasing when the diameter is 3 inches. Q Ans. — = 2.827 cu. in. per sec. 6. In Exs. 5, 7, Art. 16, is y an increasing or a decreasing function ? 3a; 4-5 Is — '^^— an increasing or a decreasing function of a; ? x + 1 7. In the Example 1, above, for what values of a is 2/ an increas- ing function of x, and for what values a decreasing function ? 8. Find where the rate of change of the ordinate of the curve y = a?—&a? + 3x-\-5, is equal to the rate of change of the slope of the curve. Ans. a;=5orl. 01? 9. When is the fraction — increasing at the same rate as a;? a? + a' Ans. When a? = a". 10. If a body fall freely from rest in a vacuum, the distance through which it falls is approximately s = 16 f, where s is in feet, and * in seconds. Eind the velocity and acceleration. What is the velocity after 1 second ? After 4 seconds ? After 10 seconds ? Ans. 32, 128, and 320 ft. per sec. CHAPTER III DIFFERENTIATION 23. The process of finding the derivative of a given function is called differentiation. The examples in the preceding chapter illus- trate the meaning of the derivative, but the elementary method of differentiation there used becomes very laborious for any but the simplest functions. Differentiation is more readily performed by means of certain general rules or formulae expressing the derivatives of the standard functions. In these formulae u and v vrill denote variable quantities, func- tions of X ; and c and n constant quantities. It is frequently convenient to write the derivative of a quantity u, — u instead of — , dx dx the symbol — denoting " derivative of." dx Thus -^^ — — — '-, the derivative of (m -|- v), may be written — (u+v). dx dx 24. Formulae for Difierentiation of Algebraic Functions. T ^—1 dx II. — = 0. dx III. '±(u + v)=^ + >h. dx dx dx 26 DIFFERENTIATION 27 dx dx dx -cT d , ^ du V . — (cm) = c — dx dx VI. du dv V u d fu\ dx dx dx\vj ir VII. -^ (w") = nit»-i*^. dx dx These formulae express the following general rules of differenti- ation : I. The derivative of a variable with respect to itself is unity. II. The derivative of a constant is zero. III. Tlie derivative of the sum of two variables is the mm of their derivatives. IV. Tlie derivative of the prodttct of two variables is the sum of the products of each variable by the derivative of the other. V. Tlie derivative of the product of a constant and a variable is the product of the constant and the derivative of the variable. VI. TJie derivative of a fraction is the derivative of the numerator multiplied by the denominator minus the derivative of the denomi- nator multiplied by the numerator, this difference being divided by the square of the denominator. VII. The derivative of any power of a variable is the product of the exponent, the power with exponent diminished by 1, and the derivative of the variable. 25. Proof of I. This follows immediately from the definition of a derivative. For, since — =1, its limit — = 1. Aa; dx 26. Proof of II. A constant is a quantity whose value does not vary. ^ Hence Ac = and — = ; therefore its limit — = 0. Aa; dx 28 DIFFERENTIAL CALCULUS 27. Proof of III. Let y = u + v, and suppose that when x re- ceives the increment Aa;, u and v receive the increments Am and Au, respectively. Then the new value of y, y + Ay = u + Au +v + Av, therefore Ay = Au + Av. Divide by Ax ; then Ay Am Av Ax Ax Ax Now suppose Aa; to diminish and approach zero, and we have for the limits of these fractions, dy _du dv dx dx dx If in this we substitute for y, u +v, we have d , , ._dttdw dx dx dx It is evident that the same proof would apply to any number of terms connected by plus or minus signs. We should then have d / , , , ^ da , dv , div , — (m ±v±w± —)= — ± — ± — - ± — . dx ax dx dx 28. Proof of IV. Let y = uv; then y + Ay = {u + An) (v + Av), and Ay = (u + Au) (v + Av) — uv = vAit, + (u + Au)Av. Divide by Ax ; ii Aw Am , / , . sAi) then -^=v h (m + Ait) — . Ax Ax Ax Now suppose Aa; to approach zero, and, noticing that the limit of M + Am is M, we have dv du , dv -2 = V \-u — ; dx dx dx , , . . d /■ ^ du , dv that IS, -— (Mv)='y— --I-M--. dx dx dx DIFFERENTIATION 29 29. Product of Several Factors. Pormula IV. may be extended to the product of three or more factors. Thus we have dx ' dx^ ' dx^ ' dx „y du , dv\ , dw \ dx dx) dx „, d\i , dv , dw = VW \-UW \-UV-—. dx dx dx It appears from the preceding that the derivative of the product of two or three factors may be obtained by multiplying the deriva- tive of each factor by all the others and adding the results. This rule applies to the product of any number of factors. To prove this, wa assume — ( MjMa ••• u„] = u^Us ••• u„^ + Ut,UsUi :■ u„ ^ H hMiM2 ••■ w„-i— • dx\ J dx dx dx Then —fihu.^ ••• m„m„+i ) = u^+^—fuiU^ •■• ■«„ ) + uiti^ ■■■ w„^^i dx\ J dx\ J dx du, , dxu , , du dx dx dx dx Thus it appears that if the rule applies to n factors, it holds also for « + l factors, and is consequently applicable to any number of factors. The derivative of the product of any number of factors is the sum of the products obtained by multiplying the derivative of each factor by all the other factors. 30 DIFFERENTIAL CALCULUS 30. Proof of V. This is a special case of IV., ^ being zero. But we may derive it independently thus : 2/ = CM, 2/ + A?/ = c(m + Am), Ay = cAm, ^ = c— , Aa; Ax' dv du (I / \ du dx dx dx dx 31. Proof of VL Lety = -- V Then , . M + Am y + Ay = ; ; v + Av therefore m + Am u vAu — uAv Ay = — — = , v + Av V {v + Av)'!) and Am Av V u — Ay Ax Ao; Ax~ (v + Av)v Now suppose Ax to approach zero, and noticing that the limit of -y + Av is V, we have du dv V- M-- dy _ dx dx dx" V- Or we may derive VI. from IV. thus : „. u Since 2/ = -) V therefore yv = u. DIFFERENTIATION 31 By IV., v^ + y^ = ^, dx dx dx „dy_du udv_ dx dx V dx' du dv , „ V u — tnerefore dy _ dx dx dx v^ 32. Proof of VII. First, suppose n to be a positive integer. Let y = u", then 2/ -I- A?/ = (m + Am)», and Ay = (m + A?<)" — M". Putting m' for M + A?«, we have Ay = u'" — «» = (m' — m) (m'"-' + u'"-^u + 'tt'"-'M^ + ••• + u"-^), that is, Ai/ = Am (m'"-' + m'"-^ u + u'^'V h ?*""'), Ax ^ 'Ax Now let Aa; diminish ; then, u being the limit of m', each of the n terms within the parenthesis becomes «""' ; therefore dy „ , du dx dx Or it may be proved by regarding this as a special case of Art. 29, where Mi, u^, ••■ and m„ are each equal to u. Then — (m") =m"-1— +m" ' ** + - to n terms dx dx dx n-idu =nu" ' — dx Second, suppose « to be a positive fraction, £. . p Let y =M', then y^ = u'; therefore -^(2/') = ^(m")- do; da; 32 DIFFERENTIAL CALCULUS But we have already shown VII. to be true when the exponent is a positive integer ; hence we may apply it to each member of this equation. This gives ^^ dx ^ dx' therefore dji^pu^du^ dx qy^-^dx p Substituting for y, u', gives dy_p M**"' dM_p J-* da dx q p_pdx q dx' which shows VII. to be true in this case also. Third, suppose n to be negative and equal to — m. Let y =«-"* = — : h VT dy _ dx _ da; _TO_idw dx u'" M^*" "~ dx Hence, VII. is true in this case also. EXAMPLES Differentiate the following functions : 1. y=zx*. ^ = ±(x*). dx dx If we apply VII., substituting u = x and n = 4, we have dx^ ' dx ^ Hence, -^ = 4*'. dx DIFFERENTIATION 33 by III., making m =3a;* and 'y = 4a!'. |(3.^)=3|(«.), byV. = 3-iaP = 12x'. Similarly, — (4a!3) =4 — (a^) = 4 •3a?'= 12a?'. dx dx Hence, ^== 12a^ + 12a;2= 12(a^ + a!^. dx 3. y = x^ + 2. dx dx dx £(a!i)=|x^ by VII. £(2)=0, by 11. Hence, -^ = -a;*. dx 2 4. y=3Vx - + -.+ a. Vx ^ ^= A(3a,i)_^(2a,-i) +A(a,-3) + ^ dx dx dx dx dx = Ix'^ -2(- ^V"^ - 3 a;-" + -r- _ 3 1 3 2a!^ a;^ '^ 34 DIFFERENTIAL CALCULUS 5. 2/-^ + ^- ^ x' + S dx dxyx' + s) Applying VI., making M = a; + 3 and v = x' + 3, we have ^ a-^ + 3-(a; + 3)2a; ^ 3-6a;-g^ Hence dy_ 3-6x-x^ 6. y = (^ + 2)i. ^ = ±(^+2)1 dx dx If we apply VII., making 2 u = 3^ + 2 and ji = ^ , we have o |(a. + 2)t = |(a.+ 2)-3^£(.^ + 2) = |(a;2 + 2)-*2a; = Hence, 3 3(x= + 2)i (*a;"~3(a,-2 + 2)^' 7. 2/ = (k2 + 1)V^ da; dte'-^ '^^ ^ ^ DIFFERENTIATION 85 If -we apply IV., making u = x' + l and -y = (x^ — x)^, we have = (af + l) ^ (x= - x)i +(a^- x)if {x' + 1). £{^-x)i=^{:^-x)-i£(x^-x)=^(x^-x)-i(3^-l). f{x' + l)=2x. dx Hence ^=hx^+l) (3 x' - 1) (a;^ - a;)"* + («?- x)^2 x ^ (g^ + 1) (3 a:^- 1) + 4 x{y?-x) ^ 1 ^ -2x^-1 2 (ari - x)^ 2 (a^ - a;)^ 8. 2/ = 3a^''-2a;« + 9^-5, ^ = 3 (10a;''-4a;= + af). 9 «-6«/;;J+ ^ 2 3 d?_^_ 6 , 2_12, 10. 2/ = (a; + 2a)(a;-a)\ || = 3Cx2-a^. 11. ,= (..-a^n i= 3,f Differentiate Example 11 also after expanding. 2a;-l dy _ 2x 12. y = (a; -1)2' dx (a;-l)» 13. 2, = a:(a^ + 5)*, | = 5 ra^ + l)(x» + 5)t 36 DIFFERENTIAL CALCULUS 14. y. ■y/a'-a^ ^ {a^-xy^ a? dx X* ie ja — x dy X dx 2 x^/ax — 3? DifEerentiate both members of the identical equations, Exs. 17-19. 17. (a;^ + ax + a%oi? -ax + a') = oi^ + aV + a\ 18. (t+^\ = 3? + 2a? + a*x-'. 19. -_^_^ = 1+ 2 1 23? — Bx + l x — 1 2x — l 20. x = t(f+aif\ f^ = {nf + a-){t^ + aFff- 01 {2t^-Sf dy^6( 3f+it)(2t'-3y ^ (f + 2y ' dt {f + 2f 22 __ Aa + 2xy dy_ 2 ax* (24 a; + 5 a)(a + 2 a;)^ ^ (2a-3x)" dx (2a-3xy'> 23. 2/=(a;+l)'(3a;-8)*(a; + 2y, ^ = 3 (13 a;^ - 24) (a; + 1)X3 x - S)\x + 2)=. „ 1 dy _ w(a!" + 1) 24. 2/ = a,(a;» + «)-' da;-(^„ + „)i- 25. 2/ = - ="--"— ^- 5^^ V2aa!-ar' <*» (2aa;-a!=)^ DIFFERENTIATION 37 26 y- (^-^)^ dy_ 8(x-a^i x" ' dx Sal" 2.-B2 + 1. 27. 2/ = fl:^i^VT^^, XT 28. x=iti-2)^j:^^ 1 = 29. y^(^—')\ (a? + a^'f dy_ dx 3 avr -a!^ dx 3 dt 4 (<^ + 1)'^ dy_ 2aV dx (a;3 + a=)^(aT^- -a')^ dy _ a;2- 1 30. y=^/?E^±J _ ^x'+x + l' da; (a;2+a;+i)Va;* + x2^1 31. 2/ = ^, -^ = -• (4a; + l)* «^ (4a; + l)* 32. y=(a2-3aa;)^(4x2 + 8aa; + 15a^)^, d^^ 4(2a;^-9a^ 33. y={x+ V«^ + l)"(n Va.-^ +!-»), _ = (rt^ _ l)(a; + Va;^ + 1)". 34. For what values of a; is 3 a;* — 8 «^ an increasing or a decreas- ing function of a; ? Ans. Increasing, wlien a; > 2 ; decreasing, when a; < 2 35. A vessel in the form of an inverted circular cone of semi- vertical angle 30°, is being filled with water at the uniform rate of one cubic foot per minute. At what rate is the surface of the water rising when the depth is 6 inches ? when 1 foot ? when 2 feet ? Ans. .76 in. ; .19 in. ; .05 in., per sec. 38 DIFFERENTIAL CALCULUS 36. The side of an equilateral triangle is increasing at the rate of 10 feet per minute, and the area at the rate of 10 square feet per second. How large is the triangle ? Ans. Side = 69.28 ft. 37. A vessel is sailing due north 20 miles per hour. Another vessel, 40 miles north of the first, is sailing due east 15 miles per hour. At what rate are they approaching each other after one hour ? After 2 hours ? Ans. Approaching 7 mi. per hr. ; separating 15 mi. per hr. When will they cease to approach each other, and what is then their distance apart ? Ans. After 1 hr. 16 min. 48 sec. Distance = 24 mi. 38. A train starts at noon from Boston, moving west, its motion being represented by s = 9 f. From Worcester, forty miles west of Boston, another train starts at the same time, moving in the same direction, its motion represented by s' = 2 «^. The quantities s, s', are in miles, and t in hours. When will the trains be nearest to- gether, and what is then their distance apart ? Ans. 3 P.M., and 13 mi. When will the accelerations be equal ? Ans. 1 hr. 30 min., p.m. 39. If a point moves so that s = V*, show that the acceleration is negative and proportional to the cube of the velocity. How is the sign of the acceleration interpreted ? 40. Given s = - + bt^; find the velocity and acceleration. 41. A body starts from the origin, and moves so that in t seconds the coordinates of its position are x = i^ + 4:f-3t, 2/ = i*!_3j2_4e. o Find the rates of increase of x and y. Also find the velocity in its path, which is ds^ jfdxV,fd^y dt \\dt ) \dt) Ans. 5f + 5. DIFFERENTIATION 89 42. Two bodies move, one on the axis of x, and the other on the axis of y, and in t minutes their distances from the origin are x = 2 ff— 6 1 feet, and j/ = 6 i — 9 feet. At what rate are they approaching each other or separating, after 1 minute ? After 3 minutes ? Ans. Approaching 2 ft. per min. ; separating 6 ft. per min. When will they be nearest together ? Ans. After 1 min. SO sec. 43. In the triangle ABC, L and M are the middle points of BC and GA respectively. A man walks along the median AL at a uni- form rate. A lamp at B oasts his shadow on the side AC Show that the velocities of the shadow at A, M, C, are as 2^ : 3" : 4^ ; and that the accelerations at these points are as 2^ : 3^ : 4?. Suggestion. — P being any position of the man, draw from L a line parallel to BP. 33. Formulae for Differentiation of Logarithmic and Exponential Functions. du VIII. |-log.M = log,e^. dx u du IX. -^l0g,M='^- dx u X. Aa» = log.a-a"^- dx dx XI. — e" = e" — • dx dx XII. ^u' = w"-'^ + log. u ■ r^ ■ dx dx dx 40 DIFFERENTIAL CALCULUS 34. Proof of VIII. Lety = log„u, tlien y + Ay = log„ (m + Am), 1 / .SI I M + Am Ay = log, (m + A«) — log„M = loga Dividing by Ax, Am ^^log/i+^r^ (1) Aa; \ u J u If Aa; approach zero, Am likewise approaches zero. u Now Lini^^ri+— r" = Lim^^ri + - j • For, if we put — = z, Am a,» (i+^Y=(i+l {'< and as Am approaches zero, z approaches infinity. But in Art. 12 we have found Lim.^fl + -j =e; therefore Lim^,^ (l-\ r" = e. Hence, if we take the limit of each member of (1), du dy 1 dx -a = log.e_- dx M DIFFERENTIATION 41 35. Proof of IX. This is a special case of VIII., when a = e. In this case log„e = log,e = l. Note. — Logarithms to base e are called Napierian logarithms. Hereafter, when no base is specified, Napierian logarithms are to be understood; that is, log u denotes log^ u. 36. Proof of X. Let y = a". Taking the logarithm of each member, we have log 2/ = M log a; dy therefore by IX., !!f = log a — ■ y dx Multiplying hy y = a", we have ^ = loga.a"*^. dx dx 37. Proof of XI. This is a special case of X., where a = e. 38. Proof of XII. ljety = u\ Taking the logarithm of each member, we have log^='ylogM; therefore by IX., ^ = _^ + log m— • y u dx Multiplying by 3/ = W, we have dx dx dx The method of proving X. and XII. by taking the logarithm of each member, may be applied to IV., VI., and VII This exercise is left to the student. 42 DIFFERENTIAL CALCULUS 1. 2/ = log(2a^ + 3a;^, 2. y = x'" log {ax + b), EXAMPLES (See note, Art. 35.) dy^ 6(x+l) dx 2a^ + 3x " a; log as' 4. y = log,„(3x+-2), 5. »='°«ISI- 6. '=-'^' 7. 1 f-t+l 8. y = a'e, 9. y = log (a'' + 6^), 10. y = (f''-iy, dx ax+b + m log (aa;+&) dy _ 1 + log X dx {x log xy dy ^ 3 logipe ^ 1.3029 dx '3x + 2 3a;+2 dy _ 2 ab dx aV - W dy 8 dx 3a;= + 10a; + 3 dy _ . 2(i=-l) ^=(l + loga)a'e*. dy _ g" log g + 6'' log 6 dx a'' + &"" dy, dx :8e^»(e^-l)'. Differentiate Ex. 10 also after expanding. 11. y = 5a; + 3 _^ dy _ 24 a; — 10 a^ _2, a;-3 12. y=(3a;-iye^-^ da; (a- - 3)= ^ = 3(9x^-1)6'^- da; ^ ' DIFFERENTIATION 43 dy _ 13. y = a^5', l^ = a;^5-(5 + a!log5) ^ dx 14. 2, = loglog«=- 1 ^^ = ^7^^^.- log X dx X (log a;)'' Differentiate both members of the identical equations, Exs. 16-18. 15. (x + e'Y = a;* + 4 cr=e' + 6 aPe^ + 4 a;e*" + e*'. 16. (a"' — e'f = a^ — 3 a^e" + 3 a'e'" — <^. 17. log (e^ + e^) = log (e'-" + e»-') +»; + «■ 18. a;'°s« = cH<>g«. in a; log a; 1 / , ^ N ^2/ log sc 20. y = log (V^ + ^+ VS), S = ^v^- 21. 2, = log(2a;+V4^^3), S^VlJ^' 22. 2/ = log ^ ^ ^ , ^ - , — —J • V^TT+1 ""^ a;Va; + l dy _ _ (x + g)^ + (a; — a)^ d^ 3(a^ - a")* 23. 2/ = a;[(loga!)2-21ogx + 2], g=(loga;) 24. y = log {Vx~+a— Vx — a), 25. y = log ( V^+3 + V^+2) + V(a; + 3)(a; + 2), ^=\/|^ _ e'" — l + e-*" dy ^ 2n (e"" — e"*") u DIFFERENTIAL CALCULUS 27. y = log^K± 1 + los + log x' #^ 1 dx X log X (1 + log x) dy _ 30 6°^ 28. ,= (3e--2e- + 2)V2e- + l, f^=-^^^—^- 29. 2/ = 3 log (V«M^ -3) + log (V?T3 + 1), ^y - ix 30. 2/ = log (a + V2 ax - «^ + da; x'~2-Vx'+3 ■y/a Va + V2 a; — a dy _ 1 dx 2{x + ^2 ax- a') 31. 2/ = log 32. y = log,(x + a), x' + l+^yx' + Sx' + l ^ dy^ ar=-l dy^ 1 da; log a; "_1 log, (a; + g) ' X + a X The following may be derived by XII. or by differentiating after taking the logarithm of each member of the given equation. £ = nx-(l + loga;). g=(aa,^)'[2 + log(ax^)]. 33. y = x'"', 34. y = (aoi?f, 35. y = 'jf^'> 36. 2/ = (log a;)", 37. 2/ = a!'"'^""' 38. 2/=(-^Y, \x-\-aJ dy ;=aa;<" +1(1 + 2 log a;). dx -^ = (n + 1) (log xyx'-'^"^ ^>"-'. dy f X Y/" 1 da; \a; + a/ \a; + -+iiog^Y a a x + aj DIFFERENTIATION 45 The method of differentiating after taking the logarithm of the expression, may often be applied with advantage to algebraic func- tions. This is sometimes called logarithmic differentiation. In this way difEerentiate Exs. 21-26, pp. 36, 37. X X 39. rind the slope of the catenary y = -(e''-\-e "), at a; = 0. "What is the abscissa of the point where the curve is inclined 45° to the axis of X ? Ans. a5 = olog. (1 + V2). 40. When does logxoa; increase at the same rate as a;? Alls. When x = logio e = .4343. When at one third the rate? Ans. When a; = 1.3029. Verify these results from logarithm tables. 41. If the space described by a point is given by s = ae'4-6e-', show that the acceleration is equal to the space passed over. 42. If a point moves so that in t seconds s = 10 log - — - feet, ^ t + 4 find the velocity and acceleration at the end of 1 second. At the end of 16 seconds. Ans. Velocity = — 2 ft., and — .5 ft. per sec. Acceleration = .4, and .025. , . 1 , o^3 9x^-36 a; + 32 43. For what values of a; is y = \o%{x — £f _ an increasing or a decreasing function ? Ans. Increasing when x > 3 ; decreasing when a; < 3. 39. Formulae for DifEerentiation of Trigonometric Functions. In the following formulae the angle m is supposed to be expressed in circular measure. d ■ „ d'"' XTTT — sinM = coSM-— • ^^^^- dx dx d ■ du XTV — COS u= —smu—-- ^^^- dx dx XV. ■^ta.nu = sec'u^- dx dx 46 DIFFERENTIAL CALCULUS d . « du XVr — cot M = — cosec"' M — ^^•^- dx dx ,_.__.__ d , du XVII. — sec M = sec M tan u -— • dx dx XVIII. — cosec M = — cosec u cot u — • dx dx XIX. -— versM=siiiM-;-' dx dx 40. Proof of XIII. Let y = sin u, then y + Ay = sm(u + ^u); therefore A «/ = sin (u + Au) — sin u. But from Trigonometry, sin ^ - sin 5 = 2 sink^ - -B) cos l(A + B). If we substitute ^4 = tt + Am, and B = u, Au\ _:„ A?t we have A?/ = 2 cos f m + -^ j sin — . Am , ^ sin— - TT Au / , Am\ 2 A?t Hence, -^ = cos m + -— ' Ax V 2 / A?t Ax Now when Aa; approaches zero, Am likewise approaches zero, and as Am is in circular measure, . Am sm — 2 Lim^„^o ^,t =1- See Art. 12 . „ dy du Hence, -^ = cosm-— • dx dx DIFFERENTIATION 47 41. Proof of XIV. This may be derived by substituting in XIII. ^-vi-)=" for u, -~u. Then A sin :^cosM = sin u(- ^\ = -sin u ^. dx \ dxj dx 42. Proof of XV. Since tan m = 5HL?£, cos u , COSM-— sinw — siuM. — cosm u,, ITT —J- dx dx by VI., ;t; tan M = • 5 H* cos u , du , . 2 du du cos^ U-— + sin'' u — — dx dx dx = sec'u- dx cos^ u cos^ u du 43. Proof of XVI. This may be derived from XV. by substitut- ing -— u for u. 2 44. Proof of XVII. Since sec u = ^ , COSM d ■ du cos M sm M — , ^-.-r d dx dx by VI., — secM = 5 = r ■' dx COS'' w cos'' u = sec u tan u—- dx 45. Proof of XVIII. This may be derived from XVII. by sub- stituting ^ — M for M. 46. Proof of XIX. This is readily obtained from XIV. by the relation ^ vers M=l — cos u. 48 DIFFERENTIAL CALCULUS EXAMPLES 1. 2/ = 3 sin 3a! cos 2x — 2 cos 3x sin 2x, -^ = 5 cos 3x cos 2a!. dx 2. y= log cos^ a; + 2a! tan « - a!^, 'T—'^^ ^^"^^ ^- 3. y = log (sec mx + tan Twa!), -^ = m sec wuc. . i/-2i7,2\ d-u 2 (a—h) tan a; 4. 2/ = log (« sin'' a! + cos'' x), — ^ = — ^^ ^ dx a tan^ x + b c 1 //I \ , /I • dv sin 6 5. y = cos a log sec (5 — a) +0 sin a, ^= . do cos (6 — a) 6. 2/=(m— 1) seC'+'a!— (m+l)sec'"~^a:,-^=(m^ — l)sec'"-^a!tan'a;. (a. -I) 7. y = log tan f aa: — ^ ], -^ = — 2a sec 2aa;. 8. r = log [sec 6 tan 6 (sec + tan eyi , ^ = (secg + tanfl/. ^ "- ^ '^ -' de tan 9. 3/ = cosec" ax cosec" bx, — = —cosec" ax cosec" 6a! (ma cot aa! + nb cot 6a!). dx ' 10. y = 2a? sin 2a! + 2a! cos 2x — sin 2a!, -^=4a? cos 2a!. da: 11. y = 2 tan' a! sec a! + tan a: sec a! — log (sec a! + tan a;), -^=8 tan^a! sec' a:, da: 1 rt sin 05 + cos X dy 2 sin x " ^ dx e 13. 2/ = e^(sin 2a! - 6 cos 2a!), ^=13e'^(sin2a;-cos2a:). 14. 2/ = log DIFFERENTIATION 49 cos X dy sin a cos (a; + o) dx cos x cos (x + a) 15. y = sin^ 4a! cos* 3x, -^ = 12 sin^ 4a! cos' 3x cos 7x. dx 1 o 1 sin a! + vers x dy 16. y = log — , -^ = sec a!. sin a; — vers x dx 17. y = (sin 2 a!)', ^ = 2/ (log sin 2a! + 2a: cot 2a:). dx \ 18. y = (tan a:)"" ', — = y (cos x log tan x + sec a;). 19. ■u = (sina:y°«'=°", -^ = w (cot a: log cos a: — tan a: log sin a:). dx 20. « = tan a: sec x + log -J-i-i^, f^ = 2sec'a:. » 1 — sin a: dx _, /i o i. \ /I dy 3sec*a! 21. 2/ = (tan a: — 3 cot a:) Vtan a:, -2 = —. "'^ 2 tan* a: sin^ie-a) ^ sin« 22. y = log-^ , -^- sin-(e + «) d^ cos a — cos tf 23. y = a log (a sin a; + & cos a:) + bx, -f-= " . da: a tan a; + 6 _si^(2^-i) dy ^' ^~~~ro i~A' f^a; H-sin4a!" sin(^2a: + |j *^^I-2 dy_ 3 25. 2/ = log- ^— ' da;~4-5sina!" 2 tan - — 1 60 DIFFERENTIAL CALCULUS no a sin x+ b vers x dy __ 2 ab vers x a sin a — 6 vers a; ' dx {a sin a; — 6 vers a;)^. In each of the following pairs of equations derive by differentia- tion each of the two equations from the other: 27. sin 2 a; = 2 sin a; cos a;, cos 2 a; = cos^a; — sin'a;. oo • o 2 tan X 28. sin 2 a; cos 2 a; = 1 + tan^a;' 1 — tan'' a; 1 + tan^a;' 29. sin 3 a; = 3 sin a; — 4 siu^a;, ■ cos 3 a; = 4 cos' a; — 3 cos x. 30. sin 4 a; = 4 sin a; cos'' a; — 4 cos a; sin' a;, cos 4 a; = 1 — 8 sin^a; cos^a;. 31. sin (m + n) a; = sin mx cos nx + cos ma; sin nx, cos (m + n) a; = cos mx cos nx — sin mx sin nx. 32. If vary uniformly, so that one revolution is made in ir sec- onds, show that the rates of increase of sin 6, when 6 = 0°, 30°, 45°, 60°, 90°, are respectively 2, V3, V2^ 1, 0, per second. 33. If is increasing uniformly, show that the rates of increase of tan 0, when = 0°, 30°, 45°, 60°, 90°, are in harmonical progres- sion. 34. For what values of 6, less than 90°, is sin 6 + cos 6 an increas- ing or a decreasing function ? Find its rate of change when 6 = 15°. Ans. ^ V2 ■ 35. The crank and connecting rod of a steam engine are 3 and 10 feet respectively, and the Crank revolves uniformly, making two revolutions per second. At what rate is the piston moving, when DIFFERENTIATION 61 the crank makes with the line of motion of the piston 0°, 45°, 90°, 135°, 180°? If a, h, X, are the three sides of the triangle, and 6 the angle opposite h, x = a COS + Vft^ — a' sin^ 6. Ans. 0, 32.38, 37.70, 20.90, 0, ft. per sec. 36. A crank OP revolves about with angular velocity w, and a connecting rod PQ is hinged to it at P, whilst Q is constrained to move in a fixed groove OX. Prove that the velocity of Q is m. OR, where R is the point in which the line QP (produced if necessary) meets a perpendicular to OX drawn through 0. 47. Inverse Trigonometric Functions. The inverse trigonometric functions are many-valued functions ; that is, for any given value of X, there are an infinite number of values of sin~' x, tan~' x, &c. For example, sin""' - = ± f ± 2mr, where n is any integer. 2 6 But if the angle is restricted to values not greater numerically than a right angle, sin"' x will have only one value for a given value of X. Then sin~' - = -, sin~' ( ) = — - . We thus regard sin~' x, 2 6 \ 2j 6 cosec"' a;, tan~' a;, and cot~^fl;, as taken between — - and -, that is, in the first or fourth quadrants. But COS"' a;, sec' a;, and vers^'a;, must be taken between and ir, that is, in the first and second quadrants, which include all values of the cosine, secant, and versine. These restrictions are assumed in the following formulae of differ- entiation. 48. Formulee for Differentiation of Inverse Trigonometric Functions. XX. — sin-'M = ax XXI. —cos-' M = dx Vl — w' dx VI -M^ du dx 62 DIFFERENTIAL CALCULUS du XXII. — tan-' M = .-^-T- dx 1 + u^ du XXIII. -^cot-'M= '^'^ dx 1+m' du XXIV. — sec-'M=- '^°" dx u^u^_i du XXV. — cosec-' u = £L dx Mt/' du XXVI. — vers-'M=- '^'^ u' do; V2«^ 49. Proof of XX. Let y = sin-' u; therefore sin y = u. By XIII., "•»l=i therefore du dy _ dx dx cos y But cos 2/= ± Vl-sin2t/=± Vl-M^ If the angle y is restricted to the first and fourth quadrants (Art. 47), cos y is positive. Hence and c.osy = dy _ = VI- du dx -u\ dx VI- u" DIFFERENTIATION 53 50. Proof of XXI. Lety = cos-'M; therefore cos y = u. By XIV., -smy^ = ^; dx dx therefore du dy _ dx dx~ sin 2/ But sin y = Vl — cos^y = Vl — u\ If the angle y is restricted to the first and second quadrants (Art. 47), sin y is positive. Hence sin 3/ = = VI - u% and dy dx' du dx VI -m' 51. Proof of XXII. Let y = tan-^M; therefore tan y = u. By XV., dx dx therefore du dy dx dx seo^y But sec^2/ = l + ta therefore du dy dx 52. Proof of XXIII. This may be derived like XXII., or from cot~^ u = tan~^ - . 54 DIFFERENTIAL CALCULUS 53. Proof of XXIV. This may be obtained from XXI. Since sec ^u= COS"' -, u dn\ 1 du du — sec-iM = — cos-'--- dx\uj u^dx dx dx dx u .U .1 J. 1 « Vm' - 1 54. Proof of XXV. This may be obtained from XX. Since cosec~'M = siu~^-j u d /1\ 1 du du (jl ^ 1 dx \uj u^ dx dx — cosec"^ M = — sin~^ , = — , = p== da; dx « /^ 1 L 1 uVv?- v-i. v-^. 55. Proof of XXVI. This may be obtained from XXI. Since vers"' u = cos"' (1 — m), d /^ s du d _. a _, /, s dx dx -vers ' M = —cos ' (1 — m) = — dx dx Vl-il-uf V2ti-u' EXAMPLES 1 , _, Bx — 1 1. « = tan ' ) " 2 ^ 3 „ . _^2a!-3 3. y = sm ' — ~ — ) i. y = vers-' (Scc^ - 8a;*), . ti = tan ' 1 x + a dy 2 dx 5x'- -2a; + l dy _ S dx a;V4 x'-9 dy_ 1 dx V(x -5)(2- ■X) dy _ 4 dx VI- -x' dy _ a DIFFERENTIATION 55 6. 2/ = tan-' (3 tan 6), ^ = — ^ ^ ae 5 -4: cos 2 6 1. y = sec-i sec= d, ^ = ^ . M Vsec^ 6 + 1 8. 2/ = yers-i-2_, ^=_^_. ^ a^ + 1 dxa^ + l 9. 2, = cot- ^:i±£l, dy ^ _Ja in _i 3a; dy ,1 10. 2/ = cosec ' ) -^ = — . a- - 1 dx X V8ic2 ^ 2a; — 1 11 , _, 3a;— 2 , ,_. 3a; — 12 dy ^ 11. y = tan'— — + cot'--— -, -| = 0. 12. y = COS"' Vvers a;, ~^~~'i Vl + sec x. 13. 2/ -a tan - - 6 tan -, Tx' (x^ + aF)(p? + h^ 14. y=cot-'^±^, ^ = -7^- bx — a dx or -{- a^ 15. 2/ = sin-i?H^^^l^2^, ^ = 1. V2 d^ ic • _iax + & dy 1 l a'—b" 16. ?/ = sin— — ;!^, --^ = , ^\h; 5- ^ 6a; + a da; 6a; + a>'l— a;^ 17. •» = tan~' (sec x + tan a;) , , = ?i- ^ V I y' da; 2 18. 2/ = sin-' , -f-= — - — -■ e' + e-'- dx e' + e " 19. 2, = cot-'(a;^-a; + l)^pot->(a;-l). ^ = ^^- 56 DIFFERENTIAL CALCULUS 20. y = t^n''i+l^^^, ^ 3 ' dx 5 + i sin 2x oi _i a; — 3 o /6 — a; dv V6 a; — o^ 21. 2/ = cos ' 2\/ , -^ = 3 * a; da; ar 22. 2/ = a;2sec-i--2VS234, ^ = 2a;sec-i-- 2 dx 2 Differentiate both members of the identical equations, Exs. 23-28. 23. 2cos-'-^p^ = cos-ia;. 24. 3 vers-^ x = vers"' [a; (2 a; - 3)^]. 25. sin"' a; + sin"' a = sin"' (a Vl — x' + a; Vl — a^. 26. tan"' ma; + tan"' nx = tan"' ^ — ' • 1 — mwar „_ _, 2a; + 2 oi _i a;+l 27. vers ' —-= 2 tan '-» — —- • x + 3 \ 2 no i. -1 a^ tan x—b^ , , /a , \ *„ _i & 28. tan ' — = tan ' -tan x ] — tan ' - • a&(l + tana;) \b J a „- oi a;'-2a; + 5,, .^x'-S dy 12a;2-20 ^^- ^ = '^°g a;- + 2a; + 5 +'^° T^' i^ a;^ + 6a;^ + 25 - , 4 a; — a;* dy _ i 30. y = tan-'j_g^^^, d^-lT^- 31. y=sin-'2('^-«^)^ gy= 2 Vg3^ . 3V3 a^a; «*» xV4 o' - a;^ 32. What value must be assigned to a so that the curve y = log, (x — 7a) + tan"' ax, may be parallel to the axis of X at the point a; = 1 ? Ans. ^ or — ^. DIFFERENTIATION 57 33. A man walks across the diameter, 200 feet, of a circular courtyard at a uniform rate of 5 feet per second. A lamp at one extremity of a diameter perpendicular to the first casts his shadow upon the circular wall. Required the velocity of the shadow along the wall, when he is at the centre ; when 20 feet from centre ; when 50 feet ; when 75 feet ; when at circumference. Ans. 10, 9j^, 8, 6|, 6 ft. per sec. 56. Relations between Certain Derivatives. It is necessary to notice the relations between certain derivatives obtained by differentiating with respect to different quantities. To express -^ in terms of — If j/ is a given function of x, then x may be regarded as a function of y. From the former relation, we have -? , and from thd latter, — These derivatives are connected dx dy by a simple relation. It is evident that — ^ = -— , Ax Ax however small the values of Aa; and Ay. As these quantities ap- proach zero, we have for the limits of the members of this equation, ^ = 1 (1) dx dx dy That is, the relation between -^ and -^ is the same as if they were ordinary fractions. For example, suppose (2) 2/ + 1" • • Differentiating with respect to y, we have dx a dy~ {y + iy By(l), J''- (^±^-- •' ^ ^ dx a a x'' by (2). 58 DIFFERENTIAL CALCULUS This is the same result as that obtained by solving (2) with refer- ence to y, giving X and differentiating this with respect to x. To express -^ in terms of -^ and — ; that is, to find the derivative dx dz dx of a function of a function. If y is a given function of z, and z a given function of x, it follows that y is a function of x. This relation may often be obtained by eliminating z between the two given equations, but -^ can be found without such elimination. dx By differentiating the two given equations, we find —and—, and 4( ciz ctoo from these derivatives, -^ may be obtained by the relation dx dy_dy^dz_ /q\ dx dz dx For it is evident that — ^ = ^ — Ao; ^z Ax however small Aa;, A?/, and Ax. By taking the limits of the members of this equation we obtain (3). That is, the relation is the same as if the derivatives were ordinary fractions. For example, suppose y = ^, z = a' — V?. (4) Differentiating these equations, the first with respect to z, and the second with respect to x, we have # = 5,4^ ^=-2x. dz dx By (3), ^ = 52^ - 2a;) = - \(ix{<£- - 'jFf, by (4). DIFFERENTIATION 59 The same result might have been obtained by eliminating z between (4), giving and differentiating this with respect to x. The relation (1) may be obtained as a special case of (3) by substituting y = x. This gives Another form of (3) is which is of frequent use. In Exs. 1-4, find — and thence ^ by (1). dy dx •' ^ ' dx dz _ dx _ -. dz dx dx dy dx _ dy dz~ dz' dx EXAMPLES (5) 1 r — "y ~^^ ^y _ (fey — Vf _ bh — aJc by — k' dx bh — ak (bx—af 2. a;=Vl + sin2,, dy ^ 2 VI + sin_y ^ 2 dx cos y v'2 — a^ 3. x= y t;y^ (l+logy)' ^ y^ l + logy' dx logy xy — ^ 4. a; = alog^^ + "+A dy Ji ^ff + ^y ^ ^ - e"'' '\fa dx a 2 In Exs. 5-8 find -^ and — , and thence -r by (3). - _ 3z ^ 2x dy ^ 12 -^"22-1' ^ 3£C-2' da; (a;+2)2 60 DIFFERENTIAL CALCULUS 6. 2/ = log— !— , 2 = 6", -^ = 2 dx e + e-" 7. y = e' + e^, 2 = log(a;-a^, ^ = 4.a^-6x' + l. dx 8. 2/ = log — ^!^, 2 = sec a; + tan a;, 62 + a dy a? — h'' dx 2 a& + {a? + b') cos x 9. Differentiate (a;^ + 2)' with respect to k'. Let ?/ = (a^ + 2)^, and 2 = ar*. It is required to find ■-^■ ^ = 4x(a^ + 2), — = 3a;l da; dx •^ ^ '^ d2 3 a:^ 3 a;" 10. Mnd the derivative of — , + -; with respect to - + -■ a" x" '^ ax Ans. 3(^^ + 1+^- \a^ x^j 11. Find the derivative of sin 3a; with respect to sin x. Ans. 3 (4 cos^ a; — 3) . 12. Find the derivative of tan~^-y^ with respect to log (1 + a;). 1 Ans. 2 Vk X 13. Find the derivative of log — -. ^t_^ with respect to a sm X— cos x . ah (a^ tan a; — 6^ cot x) a' sin'' x — y cos^ x a? + W 14. Given a; = 6 cos <^ — cos 5<^, y = 5 sin<^ — sin 5<^ ; find -^ . dx Ans. -^ = tan 3 d>. dx CHAPTER IV SUCCESSIVE DIFFERENTIATION 57. Definition. As we have seen, the derivative is the result of differentiating a given function of x. This derivative being generally also a function of x, may be again differentiated, and we thus obtain what is called the second derivative ; the result of three successive differentiations is the third derivative; and so on. Por example, if 2/ = x*, ^ = 40^, dx dxdx dx dx dx 58. Notation. The second derivative of y with respect to x, is denoted by — ^• •^ dx" That is, d^y _ d^ dy_ di? dx dx Similarly, ^_d_d^dy _ d d^y dx' dx dx dx dx dx^ d*y _ d d d dy _ d d^y da^ dx dx dx dx dx da? d"y _ d d''-'^y , dx" ~ dx dx"-^ 61 62 ' DIFFERENTIAL CALCULUS Thus, if y = a;*, ^ = 40^, dx ' ^ = 24x da;'' The successive derivatives are sometimes called the first, second, third, . . . differential coefficients. If the original function of x is denoted by f{x), its successive de- rivatives are often denoted by f'(x), /"(x), f"'(x), r{x), ... fix). 59- The nth Derivative. It is possible to express the nth derivar tive of some functions. For example, (a) From y = e"', we have dx da? daf (b) From y — = (ax + 6)"^, we have ax + b g = (-l)a(aa; + &)-, 5= (-!)( - 2)a\ax + li)-\ g = (- 1)(- 2)(- 3)a»(aa; + 6)^ = (- 1)^3 a\ax + 6)"*, dx" ^ ^L_ V -r ; (aa; + 6)"+i SUCCESSIVE DIFFERENTIATION 63 (c) From y = sin ax, we have -^ = a cos ax = a sin( ax-\- dx \ da? = a^ cosf aa; + - l = a^sinf aa;-| — ^\ ^„ = a^cos fax + — V a'^sinfax +^\ dor \2j \ 2 J — ^ = a"sin( oxH ). dx" \2J EXAMPLES 1. y = 2a?-ha^ + 20a?~5a? + 2x, ^ = 120(a^ - a; + 1). 2. y = {x'-i)i, ^ = 20(a^-l)(a^-4)i 3. 2/ = a;"* + «"" ^ = m(m - l)(m - 2)a;™-' - m(m + l)(m + 2)x-"'-\ d^y , 19 a^ 112 a!« 115 a^ 4. , = .» + .' + .^^ + .-, U = ^6+L_+L^+U_. 5. y = xnogx, g = |. 6 v-a^loRfx-l) d^^ 2(x^-3a;+3) . b. y-x'logi^x 1), ^^- ^^_^^3 7. 2/ = 4(a;-2)e==+(a;-l)e^, g = 4a!(e-4-e^)- 8. a!=(i3-3«^ + y-3)e^ ^ = 4«V. 9. r = logsec(9, ^=6sec^^-4sec='6l- 64 DIFFERENTIAL CALCULUS 10. y = e-*(ll sin 2 a; + 2 cos 5 'a;), ^=125e-'sin2a;. 11. y = tan~^ x, dV_24a;(l-a;^ da;* (1 + 3?y 12. 2, = tan-^^_^, dot? {e + 6-^f a dhj _ 2{x — a){oi? + 4 aa; + a?) d3? ix" + ay 13. y = log Vv? + d^ + tan- 14. y = (e"' 4- e-"«) sin aO, g + 4aV = 0. 15. y = a;e^(sin x — cos a;) + 3e' cos a;, — ^ = 4 aie'' cos a;, da;* 16. 2/ = e-""", f^, + (tana. l)=f = 0. da!^ ' dx 17. sin na; + cos nx d'y 2 dy 2 a dx^ X dx 18. 2/-a^(sinloga! + coslogx), x'^'^^'^ 3a;'^^ + 5« = 0. dx' dx 19. y = a^, f^, = 6''(loga)"a'^ 20. 2/ = log(3a;+2), d»v (-l)»-^3'>-l da;" (3a; + 2)» d''y_(-iy-n-3-5-(2n-3) ^'^ 2»(a; + l)'-i 21. 2/ = Va; + 1, 22. 2/= sin 5 a; sin 2 a;, — ^= .1 '2 ~3-cos{dx+'^\- 7"cos('7a;+^y SUCCESSIVE DIFFERENTIATION 65 The following fractions should be separated into partial fractions before differentiating. dx" 2 *!^=(_l)"|n 23. y-^-X^ 21 V- 3=«-4 ^ 2ar' + 3a;-2 25. 7- ^^ ^ &x^-5x-Q' d^y t = (-ly\n dx' 26 ^^ ^a^ + a' + l ^i , 2a! + 2 • ^ 2a^-a;-l 2x''-x-l' ^=(-l)"k 1 . 1 (^ - 1)»+1 (a; + !)«+!_ 2 2» {X + 2)"+i (2 X -!)»+' 2"+i 3»+i _(2a;-3)"+i (3a;4-2)»+' 9n+l 3(a!-l)"+i 3(2x+l)»+i_ 27. 2/ = 28. 2/ = aK + lV ax — ll' dx" (x + 2)"+2 (^»y _ 4(-l)"a"|w(aa; + n) d^~ (ax - 1)"+^ 60. Leibnitz's Theorem. This is a formula for the nth derivative of the product of two factors in terms of the successive derivatives of those factors. A special case of Leibnitz's Theorem, when n=l, is Formula IV., d , s du , dv ■ — (uv) = — v + u-—- dx dx dx For convenience let us use the following abridged notation : d^v (1) dv d?v du dhi dx" d"u dx" 66 DIFFERENTIAL CALCULUS Then (1) becomes -—(uv) = ujv+uvi (2) dx DiflEerentiating (2), dp (uv) = U2V + VrjVi + tti^i + UV2 = U2V + 2 Mi^i + UV2, (for cP —- (uv) = UgV + U2V1 + 2 u^Vi + 21I1V2 + U1V2 + uvs = U3V + 3 U2V1 + 3uiV2 + UV3. We shall find that this law of the terms applies, however far we continue the differentiation, the coefficients being those of the Bino- mial Theorem ; so that — (uv) = u„v + nu„.iVi + -=^--- — ^M„_2ti2 H h nuiv„_i + uv„. (3) This may be proved by induction, by showing that, if true for — (uv), it is also true for (uv). This exercise is left for the student. In the ordinary notation (3) becomes rf" fayw'^"",. I ^d'^-'ndv n(n-l)d^-H d'v dx"^ ' die" da;"-ida; [2 dx-'-^da^'; , dud"~''-v , d"v -\-n ; + M — • dx dx"--^ dx"" EXAMPLES 1. Given « = a^sin2a;: find by Leibnitz's Theorem —^ d* dx'^ From (3), -— (uv) = u^v + 4 u^v^ + 6 U2V2 + 4 u^v^ -{■ uv^. U = 3?, Wi = 3 X^, U2 = &X, M3 = 6, Ui = 0. V = sin 2x, Vi = 2 cos 2x, V2= —i sin 2x, Vs= —8 cos 2 x, Vi = 16 sin 2 X. SUCCESSIVE DIFFERENTIATION 67 ^/ = ^ (a!^ sin 2 k) = 0- sin 2 a; + 4-6. 2 cos 2 a; + 6-6 M (- 4 sin 2 a;) da;* dx'^ ^ -r v >» + 4 • 3 a.-^ (- 8 cos 2 a;) +a^ 16 sin 2 a; = 16 [(a^ - 9 a;) sin 2 a; + (3 - 6 a;2) cos 2 a;]. 2. Given. V =*e'"; find — • Here M = e'", Ui = ae'", — ?«„_i = a»-'e'", M„ = a"e'". ?; = a;, '"1 = 1, ^2 = 0, ^3 = 0, •■•. Substituting in (3), we have ^ = — (e'^x) = a'e^a; 4- na^-^e"^ =ctf^^e"(ax + n). 3. y^ix+iyV^, g^ 3(5^-14. + 13) . 6. 2/ = sin a; log COS a;, ^ = sina![logcosa! — 2tan«a!(3tan«fl!+ 5)]. QiiC 7. v^se'a'', ^ = a'(logffl)"^«[(« log a + »)«-»]. da;" •*• ^~(a;.+ l)^' da;» '^ ^ ^ (a; + l)''+'' CHAPTER V DIFFERENTIALS. INFINITESIMALS 61. The derivative -^ has been defined, not as a fraction having a dx numerator and denominator, but as a single symbol representing the limiting value of — ^, as Aa; approaches zero. In other words, the derivative has not been defined as a ratio, but as the limit of a ratio. We have seen (Art. 56) that derivatives have certain properties of fractions, and there are some advantages in treating them as such, thus regarding -^ as the ratio between dy and dx. dx Various definitions have been given for dx and dy, but however defined, they are called differentials of x and y respectively. The symbol d before any quantity is read " differential of." 62. Definition of Differential. One definition is the following: The differential of any variable quantity is an infinitely small in- crement in that quantity. That is, dx is an infinitely small Ax, and dy an infinitely small Ay. By the direct process (Art. 16) of finding the derivative of an algebraic function, Ay is generally expressed in a series of ascending powers of Ax, beginning with the first. For example, \i y = a?, y + Ay=ix + Aa;)', and Ay = ^oi?Ax + ^x{Axy + {Axf. ... (1) In finding the derivative we have ^ = Ba?+ZxAx + (AxY, Ax ^ J ' in which, as Ax approaches zero, the second member approaches Za? as its limit, the second and third terms approaching the limit zero. 68 DIFFERENTIALS 69 If we let Aa; approach zero in equation (1), every term approaches zero, but there is nevertheless a marked distinction between them, in that the second and third terms, containing powers of Ao; higher than the first, diminish more rapidly than that term. Thus we have ' A?/ = Sx^Ax approximately, and the closeness of the approximation increases as Aa; approaches zero. From this point of view, regarding dx and dy as infinitely small increments, we may write dy = Sx^dx, not in the sense that both sides ultimately vanish, but in the sense that the ratio of the two sides approaches unity. Thus dy=3a^dx, and ^ = 3a;=, dx are two modes of expressing the same relation. According to the first, An infinitely small increment of y is Zv? times the corresponding infi- nitely small increment of x. According to the second. The limit of the ratio of the increment of y to that of x, as the latter increment approaches zero, is Saf. Just as we sometimes say " An infinitely small arc is equal to its chord," instead of "The limit of the ratio between an arc and its (?hord, as these quantities approach zero, is unity." So in general, if y =/(»), Um^,^,^^f'(x), that is, ^=f'(.^) + ^, ' Ax where e approaches zero as Ax approaches zero. 70 DIFFERENTIAL CALCULUS Hence i^y = f (x) £^x + € Ajc, and as the term eAa; diminishes more rapidly than the term f'(x)Ax, we have Ay =f'(x) Ax approxim ately , or dy=f'{x)dx. Corresponding to every equation involving differentials, there is another equation involving derivatives expressing the same relation, and the former may be used as a convenient substitute for the more rigorous statement of the latter. Thus the use of differentials is not indispensable, but convenient. It should always be kept in mind that their ratio only is important, the derivative being the real subject of mathematical reasoning. 63. Another Definition of Differentials. The differentials dy, dx, are sometimes defined as any two quantities whose ratio equals the derivative dy dx : tan RPT. Y Q/^ F ^ T R ' dx y X Let us see what this defini- tion means geometrically. If we regard the derivative as the slope of a curve, dy dx By this definition of differen- tials, dx may' be any distance PE taken as the increment of a^, and dy is then RT, the corre- sponding increment of the ordi- nate of the tangent line at P- That the two definitions are consistent will appear, if we diminished. The smaller we take PR, the more nearly is ^^ equal to unity, or RQ in other words, the more nearly is RT equal to RQ. If PR is supposed to be infinitely small, this definition of differ- entials becomes that of the preceding article. suppose PR to be indefinitely RT DIFFERENTIALS 71 The second may be said to be the more rigorous of the two defini- tions, but the first has the advantage of being more symmetrical, and better adapted to the various applications of the calculus to mechanics and physics. 64. FormulEB for Differentials. The formulae for differentiation may be expressed in the form of differentials by omitting dx in each member. To each of the formulae for a derivative, corresponds a formula for a differential. Thus we have II. dc = 0. III. d(u + v) = du + dv. IV. d(uv) = vdu + udv. VI. ify\ vdu — udv \y) v^ VII. d{u") = nu"~^ du. IX. , , du d log u = — u XI. de" = e" du. XIII. d sin u = cos u du. XIV. d cosu= — sin u du. XV. d tan u = sec^ u du. XVI. d cot (( = — cosec^ u du. XVII. d sec u = sec u tan u du. XVIII. d cosec M = — cosec u cot u du, XX. J ■ 1 du VI -m' XXII. , , , du d tan u = - :• 1+u^ XXIV. du u^u' - 1 XXVI. , , du d vers-' u = • V2m-w' 72 DIFFERENTIAL CALCULUS Differentiation by the new formulae is substantially the same as by the old, differing only in using the symbol d instead of — . ■ For example, let y — dy = d(- x' + Z ■a; + 3\ . (a!^+3)d(a! + 3)-(a; + 3)d(a^+3) ^ (a;^ + 3) da; - (a; + 3) 2 xdx ix'^ + Zf ^ {oi? + ^-2x^-&x)dx ^ {B-&x-x^dx {x^ + Sy {x' + 2,f If we wish to express the result as the derivative, we have only to divide by dx, giving dy __Z — Q X —x^ dx {x' + ^f EXAM PLES Differentiate the following functions, using differentials in the process : 1. y = {x-l)(2-Sx){2x + S), dy = {-19, x' + 2 X + 11) dx. (i + l)(f + 2)' (tJ,l)\t + 2y 3. y = V^^ VW^2, # - vmVF2 ^" 4. r=?iH!_^, ^,.^(2 + sin^e)sing^g^ cos'' 9 cos* ^ 5. 2/=e'(a;3-6a!^ + 24a;-40), dy = e^ {^ + 'l\dx. INFINITESIMALS 73 6. r = sin 6 log tan 6, dr = cos (9 log tan ed6 + sec 6 dO. 7. 2/ = tan-'-^. J 4 cZa; 4-x^' "^-4T^ 8. 2/ = sin-i 3a; + SxVl - 9 a^, d?/ = 6 Vl - 9 a;^ da;. 9. ,^ = tan-Jtan3e, dr2b''xdx = % giving ^ = — , as before. da; a'y In deriving —^;—K' •■■> derivatives should be used rather than differentials. '^'^ "^^ EXAMPLES Find the following derivatives. 1. (a;-a/ + (2/-6)2 = c^ dy__ x — a d^ _ _ c^ d^y _ _ Zc^(x — a) dx y — b'dv? (y— by' dx'' (y — 6)^ • 2. x=y log (xy), -^= '"~y . dx xy -\- x^ 3. (cose)*=(sin,^)^ g^^loSsin.^+<^tang_ ^ ^ de log cos 6-61 cot <^ 4. ax'+2hxy+by^=l, dy __ ax + hy d^y _ h- — ab d^x _ h^—ab dx hx + by dx? (Jix + byy dy' {ax + hyy IMPLICIT FUNCTIONS 77 5. aa?+2hxy + bf = 0, ^ = 1, dx X ^ 'M sin2e'de' siii229 7 „_2r-ra--7/Mos- (-r-wl dy _ 2y-x d^y _ (x- yf 7.2/ ^a;_(a, 2/)log(.<; y), ^^- ^ ,^- ^ ■ 8. {5x + y + &)\Sy-Sx + i)=c, f = ^^^- 9. r=sin20 + 2r + l = O, f^Y+2)-cot 6^= r^ \ddj do 10 e^» = a«&» ^^_ ?/-loga ^ d^y _ 2 (y- logo) _ ' dx X — log 6 daf {x — log &)^ CHAPTER VII SERIES. POWER SERIES 67. Convergent and Divergent Series. The series «1 + M2 + M3+ ••■ +?(„ + M„+i, (1) composed of an indefinite number of terms following each other according to some law, is said to be convergent when the sum of the terms approaches a finite limit, as the number of terms is indefi- nitely increased. But when this sum does not approach a finite limit, the series is divergent. That is, if S^ denote the svim of the first n terms of (1), the series is convergent, when Lim„^„ S„ = some definite finite quantity. When this condition is not satisfied, the series is divergent. Thus the geometrical series, a + ar+ar' + ar'+ •■■ is convergent when r is numerically less than unity, and divergent when r is numerically greater than unity. For S„ = a + ar + ar'+ ■■■ +a»-"-^ = "^^^ ~^) . 1 —r When *|r|l, Lim„^„ (S,, =oo. When I r I =1, the series is also divergent. 68. Series of Positive and Negative Terms. Absolute and Conditional Convergence. In the case of series composed of both positive and negative terms, a distinction is made between absolute convergence and conditipnal convergence. * I r I denotes the numerical value of r. 78 SERIES 79 Before defining these terms, the following theorem should be noticed: A series whose terms have different signs is convergent if the series formed by taking the absolute values of the terms of the given series is convergent. Without giving a rigorous proof of the theorem, we may regard the given series as the diiferenee betv/een two series formed of the positive and negative terms respectively. The theorem is then equivalent to this : If the sum of two series is convergent, their difference is also convergent. A series is said to be absolutely convergent, when the series of the absolute values of its terms is convergent. A series whose terms have different signs may be convergent without being absolutely convergent. Such a series is said to be conditionally convergent. For example : 1 \-- \- (1) converges to the limit log^ 2, but it is not absolutely convergent, since is divergent (see Art. 70). Series (1) is accordingly conditionally convergent. Ill But ^~"2^'^'5^~I^^"' is absolutely convergent (see Art. 70). 69- Tests for Convergence. The following are some of the most useful tests. In every convergent series the nth term must approach zero as a limit, as n is indefinitely increased. That is, the series Mj + m, + M3 -I + m„ H is convergent, only when Lim,,^ ?/„ = 0. For S^ = *S'„_i + u„. 80 DIFFERENTIAL CALCULUS If the sum of the series has a definite limit, Lim„^„ S„ — Lim„^„ S„. j. Hence Lim„^„M„ = (1) Por a decreasing series whose terms are alternately positive and negative, this condition is sufficient.* For example, l — i + i-f-i... ^ ' 2 3 4 is convergent. But the decreasing series 2_3 4_5 1 2 3 i" is divergent, as it does not satisfy (1), since Lim„^„ m„ = 1. The^sum of this series oscillates between two limits, log, 2 and 1 + log, 2, according as the number of terms is even or odd. Such a series is called an oscillating series. For a series whose terms have the same sign, the condition (1) is not suf&cieilt. For example, the harmonic series 1 + - + -+- + -- 2 3 4 is divergent (see Art. 70). 70. Comparison Test. We may often determine whether a given series of positive terms is convergent or divergent, by comparing its terms with those of another series known to be convergent or diver- gent. In this way the harmonic series '+M+l+hhhh-- ■ ■ ■ w may be shown to be divergent, by comparing it with * The proof of this is omitted. SERIES 81 Each term of (1) is equal to, or greater than, the corresponding term of (2). Hence if (2) is divergent, (1) is also divergent. But (2) may be written 1 + 1+2 + 4 _8_^... -'- + 1 + -2 + 1 + -2 + - The sum of this series is unlimited ; hence (2) is divergent, and therefore (1). Consider now the more general series i^ + 2^ + 3? + ^+"' ^^^ lip = l, the series (3) becomes (1), which is divergent. If p < 1, every term of (3) after the first is greater than the cor- responding term of (1). Hence (3) is divergent in this case also. If p > 1, compare L + L+L + L + L + L+L + L + ... + i_ + ... . (4) 1? ^ 2" 3" 4" 5" 6" 7" 8P 15" ^ -' Every term of (4) is equal to, or less than, the corresponding term of (5). But (5) may be written L+2_ 1 8_ ip ^ 2'' 4^ 8" ' 2 a geometrical series whose ratio, — , is less than unity. Hence by Art. 67, (5) is convergent and consequently (4). Thus it has been shown that when p ^1, the series (3) is divergent; when p >1, the series (3) is convergent. The series (3) together with the geometrical series are standard series, with which others may often be compared. 82 DIFFERENTIAL CALCULUS 71. Cauchy's Ratio Test. This depends upon the ratio of any term to the preceding term. In the series .... (1) «1 + "2 + Ms H 1- Wn + W«+l+ ■ this ratio is Let us first consider, from this point of view, the geometrical series a + ar + ar^+-+ar'' + ar+'+- (2) Here the ratio -^ = r, and is the same for any two adjacent terms. We have seen (Art. 67) that this series is convergent or divergent, according as i i ^ ^ i i ^ i ' ^ |r|< 1, or ]?-|> 1. That is, (2) is convergent or divergent according as < 1, or >1. If now (1) is any series other than the geometrical series, the ratio -^ is not constant, but a function of n. The series is then convergent or divergent, according as Lim„ < 1, or Lim„, >1. (3) Let We will first suppose (1) to be a series of positive terms. Lim„ = p. Suppose p < 1. By taking n sufficiently large we can make — approach its limit p as nearly as we please. There must be some value m, of n, such that when n ^ m, -^ < r, a proper fraction. Hence m„+i < uj-, u„+2 < "m+i'' < u„r^, etc. W» + Mm+1+ W„^2+ ••• 1. By similar reasoning, when n ^ m, Hence -^ > r, an improper fraction. ™+i > «',„»*, M»+2 > «»+!'• > uy+ etc. Since r > 1, the second member, and therefore the first member, must be divergent. Thus the theorem is proved for a series of positive terms. If the terms of (1) have different signs, it is evident from Art. 68 that the series will be absolutely convergent if Lim„ <1. It is also true that for different signs, (1) will be divergent if Lim„ >1. The proof of this latter statement is omitted. If Lim. = 1, the series may be either convergent or divergent. There are other tests for such cases, but they will not be considered here. EXAMPLES 1. Is the following series convergent? J_ + J_+J^ + ... + . 1-2 2-22 3-23^ '> Applying (3), Art. 71, we have Lim n + • 2 (« + 1) 1 2 (n + 1) 2 As this is less than unity, the given series is convergent. Its limit is log^ 2, as will appear later. 84 DIFFERENTIAL CALCULUS Determine which of the following series are convergent, and which divergent. 2. - + -+,- + ,-+•••• By(3), Art.71. ^ II 11 ll 3. l+^+lL+i_+.... By (3), Art. 71. 4. 1 + I + I+I + ---. By (1), Art. 70. o o 7 ■ 1-2 2-3 3-4 4-5'^ ' 6- 1 + I+I + I + I+-. «• |-i + f-| + n--- By(l),Art.69. 9- o'^S'^io'' ^ 2 I i "! • Compare with (3), Art. 70. 10. -^ -^ p+---. Compare with (3), Art. 70. 1+Vl 1 + V2 1 + V3 ^ ^" 11. log?_log| + log|-log^+..-. By (1), Art. 69. 12. sec ^ - sec - + sec ^ - sec ^ + • 3 4 6 6 13. sin=^+sin2^ + sin2^ + sin2| + . 2 3 4 5 14. 15. 16. 1 + 1 r+1 2^+1 ' 3^ + 1 42 + 1 2+1 , 3 + 1 POWER SERIES 4+1 85 + • 1 + 1 2 + 1 3 + 1 4 + 1 . 12 + 1 2^+1 32 + 1 4^' + l '+^-+l±^+^+l^+ -1 3''-l 43-1 5=-l Exs. 2, 5, 6, 9, 11, 13, 14, 16, convergent. Exs. 3, 4, 7, 8, 10, 12, 16, divergent. Exs. 8, 12, oscillating. 72. Power Series. A series of terms containing the positive in- tegral powers of a variable x, arranged in ascending order, as fto + ttjo; + a^^ + a^ -j , is called a power series in x. The quantities Oo^ ^i) (h> supposed to be independent of x. Eor example, l + 2a; + 3a--^ + 4a^H , are 1-^ , ^' ^ [6 t-t + are power series in x and 3/ respectively. 73. Convergence of Power Series. A power series is generally convergent for certain values of the variable and divergent for others. If we apply the ratio test, (3), Art. 71, to the power series ao+ajX + a^-\ han^-H , (1) we have for the ratio between two terms Lim„ w„+, _a. X _ M„ a„_i M»+l = Lim,^ a^x = |a;|Lim„=„ a„ M„ a„-i a„_i 86 DIFFERENTIAL CALCULUS The series (1) is convergent or divergent according as that is, according as |a;|Lim„ >i; a" requires further examination. For example, consider the series l + 2x + 3a^ + ia^-\ h nx"-' + (n + !)«;" + ••■. Here a„ n + 1 Lim, n + 1 = 1. (2) Hence (2) is convergent or divergent, according as \x\ < 1 or \x\ > 1. We may say that (2) is convergent when — 1 < a; < 1, and the in- terval from — 1 to + 1 is called the interval of convergence. EXAMPLES Determine the values of the variable for which the following series are convergent : 1. l + x-\-a? + 3^+--. 2. ^ 4--^ L.^!_+.... 1-2 2-3 3-4 /yi2 ™i3 nA 3. a;+-+- + - + .... 2 3 4 . y? , x" x^ , *• "-3 + 5-7+-- POWER SERIES 87 0. x-\ — • — 2 3 2.45 2.4.67 6. l + a;-t- — + — + - f ••■• 7 1_ — +^_^ + .. |2 ' |4 [6 [3+ [6 |7 + Answers Exs. 1-5, convergent when — 1 < a; < 1. Exs. 6-8, convergent for all values of x. CHAPTER VIII EXPANSION OF FUNCTIONS 74. When, by any process a given function of a variable is expressed as a power series in that variable, the function is said to be expanded into such series. Thus by ordinary division ^— = l-a! + 9^-af'+- (1) 1 + x ^ ^ By the Binomial Theorem (a; + ay — a* + 4: a^x+ 6 aV + i aa? + »*. (l-x)-^=l + 2a; + 3a^ + 4a;2H . . . . (2) The methods employed in these expansions are applicable only to functions of a certain kind. We are now about to consider a more general method of expansion, of which the foregoing are only special cases. It should be noticed that when a function is expanded into a power series of an unlimited number of terms, as (1) and (2), the expansion is valid only for values of x that make the series con- vergent. For such values, the limit of the sum of the series is the given function, to which we can approximate as closely as we please by taking a sufficient number of terms. The general method of expansion is known as Taylor's Theorem and as Maclaurin's Theorem. These two theorems are so connected that either may be regarded as involving the other. We shall first consider Maclaurin's Theorem. 88 EXPANSION OF FUNCTIONS 89 75. Maclaurin's Theorem. This is a theorem by which a function of X may be expanded into a power series in x. It may be expressed as follows : fix) =/(0) +/'(0)5+/' (0)|+/"(0)|+ ..., in which f(x) is the given function to be expanded, and/' («),/" (x), f"{x),---, its successive derivatives. /(0);/'(0)!/"(^)) •"> *s the notation implies, denote the values of f{x), f (x), f (x), -, when a; = 0. 76. Derivation of Maclaurin's Theorem. If we assume the possibility of the expansion of f(x) into a power series in x, we may determine the series in the following manner : A ^mim p f(x)=A + Bx+Ox' + Dai' + Ex*+-, ... (1) where A, B, C, ■•• are supposed to be constant coefficients. Differentiating successively, and using the notation just defined, we have f'{x) = B + 2Cx + 3Da^ + 4:EaP+- .... (2) f"{x) = 2C+2.3i)x + 3-4:Eaf- (3) f"'(x) = 2-3D + 2-3-iEx + — (4) f-{x) = 2-3-4.E+- (5) Now since equation (1), and consequently (2), (3), ••• are supposed true for all values of x, they will be true when a; = 0. Substituting zero for x in these equations, we have from (1), AO)=A, A=f{0), from (2), f(P) = B, B=f{0), from (3), /"(0) = 2(7, C=-C^, 90 DIFFERENTIAL CALCULUS from (4), /'"(O) = 2 . 3 D, D = -f^^^, from (5), f^{0) = 2-3-4LE, ^ = CT1, Substituting these values of A, B, C, ■■■ in (1), we have m =/(0) +/'(0)f +/"(0)||+/"'(0)|. X , ^,/r,s^ , v^lfl/ma;^ I ... (6) 77. As an example in the application of Maclaurin's Theorem, let it be required to expand log (1 + x) into a power series in x. f(x) = log(l + X), /(O) = log 1 = 0. /(«') = T^ = a + a')-S /'(0)=1. f"(x)=-(l + x)-^ /"(0)=-l. f"{x)=2{l + x)-', /"'(0)=2. /" (a;) = - L3 (1 + x)-*, r(0) = - [3. r(x)=\i(i+x)-\ r(0)=ii. Substituting in (6), Art. 76, we have x' 2a? [3a* [4£c' log(l+a;)=0 + l.a;-1.2+^-^+-^---- ™2 /yi8 rtt4 ™o log(l + c«) = a--+--- + --.... 78. If in the application of Maclaurin's Theorem to a given function, any of the quantities, /(O), /'(O), /"(O), ••• are infinite, this function does not admit of expansion in the proposed power series in x. In this case fix) or some of its derivatives are discontinuous for a; = 0, and the conditions for Maclaurin's Theorem are not satisfied (see Art. 94). The functions logs;, cot a;, aj¥, illustrate this-case. EXPANSION OF FUNCTIONS 9l EXAMPLES Expand the following functions into power series by Maclaurin's Theorem : af a? X* 1. e"' = l + a; + — + — + — H . Convergent for all values of x. \±. I£ \L /VIM rtflO /yt' 2. sina; = a; — — + — — r— H • Convergent for all values of x. [3 [5 [7_ 3 cosa; = 1— — + — — — 4-... Convergent for all values of x. [2^[4 [6_ • 4. (a + a;)» = a» + na"-'a; + ^^^^^^-=^a'-V _^_ n{n-l)(n-2) ^^^^ _^ _ Convergent when |a!| < a. 12. 5. log„(l+a;)=log.e(^a!-|+|-j+-). Convergent when la;l< 1. 6. log (l-a;) = -a;------ . Convergent when | a; |<1. ^ o 4 7. tan-'a;= a; -- + ---+•••. Convergent when ja;) <1. 3 5 7 Here / (x) = tan"' x, /' (a;) = -1- = 1 - a!^4- «'-»"+•• •, f"{x) = -2a; + 4k=- 6a=+ -, Q • 1 ,1 »^_l1-3 a^, 1-3-5 a;' , Convergent when |a;| <1. 92 DIFFERENTIAL CALCULUS Here f{x) = siii~^a;, /'(^)=-=L==(i-xr^. VI — ar Expanding by the Binomial Theorem, /' {x) = 1 4-aa^+ bx*+ ca^-\ , , 1,1-3 1-3-5 where a = -, b = —,c = ^-^, -^ f"(x) = 2ax + Aba^+6ca:^+--, 9. sin (x + a) = sin a + x cos a — r^ sin o, — — cos a - [2 \3 Convergent for all values of x. 10. log(l + a; + aO = a' + f-^ + f + |--. Convergent when | a; | < 1. 11. e' sinx=:x + af-\ . Convergent for all values of x. 12. e' cos x = l+x — -I . Convergent for all values of x. 3 6 13. tana; = x + — +— — H . 3 15 14. sec a; = l + - + -—-+•". 2 24 dj £C dC 15. logseca= = - + - + -+.... Defining the hyperbolic sine, cosine, and tangent by _ e"— e~ ' e'+e- sinha; = — ^!^ — , cosh a; = — ^ — , tanha; = — , show that 16. sinh a;=a; + — + — + ■ \3 [5 EXPANSION OF FUNCTIONS 93 17. cosha; = l + ^ + ^+. \2'\i 18. tanha! = x-- + — ■ 3 15 19. Show by means of the expansions of Exs. 1, 2, 3, that «» vZ-l _ cos X + V— 1 sin X, W-i — = cos X - W~i sina;. These are important relations. 79. Huyghens's Approximate Length of a Circular Arc. If s denote tlie length of the arc AGB, a its chord, and h the chord of half the arc, it may be shown that \h-a s =- -, approximately. Let ' 4[6 + ...W3S 1 4|5+ ; If s is an arc of 30°, <^ = -^, and the error < ^^^^ • If s is an arc of 60°, ^3 3-33 5-3= 7-3' J Four terms of this series give log 2 = .6931. The computation may be arranged as follows: 5 = .333333 o i = .037037 3' 3 1 3-3«" = .333333 .= .012346 •^ = .004115 3= = .000457 = .000051 5-3= .000823 1 7-3' .000066 1 9-3'* .000006 .34657 2 .69314 The numbers in the first column may be obtained by dividing suc- cessively by 9. l + x 1+-. Any number may be put in the form ^ and log 3 = may be found like log, 2. 96 DIFFERENTIAL CALCULUS But having log 2,. it is easier to compute 1 + i 1 3 1 5 log - = log -, '-I 3 and then log 3 = log - + log 2. Li Let the student make this computation. Find log 5 from log - = log • 4 In a similar way find log 7 from log 5. Having obtained the logarithms of 2, 3, 5, 7, find the other loga- rithms in the table at the beginning of this article. To obtain the common logarithm, that is, logarithnijo, it is only necessary to multiply the Napierian logarithm by .4343, the modulus of the common system. Find thus the common logarithms of the numbers in the foregoing tables, — first, of 2, 3, 6, 7, and from these the others. 82. Computation of ir. From Ex. 7, p. 91, by letting a; = 1, we have a slowly converging series. To obtain a series converging more rapidly, we may use tan"^ 1 = tan~^ - + tan ~' -, Z o from which t = ;^— ;^— ;^ + i^-?r< — ;;— ?;^-l 4 2 3 • 2^ 5 • 2* 7-2 3 3-3= 5-3= 7-3' EXPANSION OF FUNCTIONS 97 By taking 9 terms of the first series and 5 of the second, the student will find 1=0.463647... + 0.321751 ..• and IT = 3.14159".. Other forms of tan~^ 1 may be used, giving series converging even more rapidly, as tan-i 1=2 tan-i i + tan-'- . 3 7 tan-' 1 = 4 tan-' tan-i 5 239 By these formulae the computation has been carried to 200 deci- mal places. 83. Taylor's Theorem. This is a theorem for expanding a function of the sum of two quantities into a power series in one of these quantities. As the Binomial Theorem expands (a; + hy into a power series in h, so Taylor's Theorem expands f{x + h) into such a series. It may be expressed as follows : f{x+h)=f{x)+r(x) h +/"(x)j| +/"' ix)^+ .... 84. The proof of Taylor's Theorem depends upon the following principle : If we differentiate f(x + h) with respect to x, regarding h con- stant, the result is the same as if we differentiate it with respect to h, regarding x constant. That is, ^f{'« + K)=-^f(^ + '0- dx dh For, let z = x + h, (1) 98 DIFFERENTIAL CALCULUS then by (3), Art. 56, dx dx dz dx A /(a; + h)= —f(z) =^f(z)—- dh-'^ ' dV^' dz-'^ 'dh But from (1), — = 1, and — = 1 : ^ " dx ' dh ' therefore — f(x + li) = — f(x + h). 85. Derivation of Taylor's Theorem. If we assume the possibility of the expansion of f(x + h) into a power series in h, we may deter- mine the series by the aid of the preceding article. Assume f{x+h)=A + Bh+Ch'' + DJv'+ •■■ (1) where A, B, C, ••■ are supposed to be functions of x but not of h. Differentiating (1), first with respect to x, then with respect to h, d J., , ,s dA , dB, , dCjo , dDjn , dx dx dx dx dx ^f(x + h) =B + 2Ch + 3J)h^+-. By Art. 84, the first members of these two equations are equal to each other, therefore ^ + i^h + ^h'+... = B + 2Ch + 3Dh^ + -^. dx dx dx Equating the coefficients of like powers of h according to the principle of Undetermined Coefficients, we have dA -r, -r, dA dx dx dB^2C 0=- — dx ' 2 da?' dx [3 dx> EXPANSION OF FUNCTIONS 99 The eoeflScient A may be found from (1) by putting h = 0, as that equation is supposed true for all values of h. Then A =/(«). Hence £ = ^ = f'(x). Substituting these expressions for A, B, C, ■•• in (1), we have f(x + h)=f{x)+f{x)h+f<'{x)^+f"'(x)^+:.. . . (2) 86. Maclaurin's Theorem may be obtained from Taylor's Theorem by substituting a; = 0. We then have fih) =/(0) +/'(0) h +/"(0) ^ +/"'(0) ^ + -. This is Maclaurin's Theorem expressed in terms of h instead of x. 87. As an example in the i application of Taylor's Theorem, let it be required to expand sin (x + h) into a power series in h. fix + K) = sin {x + h); hence f{x) = sin x, /'(») = cos X, f"(x) = — sin X, f"(x) = — cos x, /•^(a;) = sin x. Substituting these expressions in (2), Art. 86, we find sin (x + h) = sin a; + A cos a; — — smw — — cos a; + — sin a; H . \2_ \6_ |4_ 100 DIFFERENTIAL CALCULUS EXAMPLES Derive the following expansions by Taylor's Theorem : 1. cos (x + h) = cos as — h sin a; — -t^ cos a; + — sin a; + •••. \1 If 3. (x + hy = x' + 73^h+--: 5. log(a: + ;0=loga; + --2^ + 3^-— , + -. 6. tan (x+h) = tan x + h sec^ x + h^ sec^ a;tan x 7j3 + -5-(3sec^flj — 2sec^a;) H — . o 7. Compute from Ex. 1, cos 62° = 0.4695. 8. Compute from Ex. 6, tan 44° = 0.9657, tan 46° = 1.0355. 9. /(^ + h) +f(x-h) ^^^^^ +!/>>(.)+ g^yw(,) + .... ^ [± li 10. /^^±^i^/(^^ =/./'(.) + 1 r'(.)+^r (.)+•- As a special case of Ex. 10, derive llog^±/* = ^ + ^ + -^+.... 2 a; — /i. X 3a^ 5oc^ 11. /(2a;) = /(x)+x/'(a;) + -|/"(a,) + ^/'"(a;) + .... ■^l^l + a;j -^ ^ ^ 1 + a;-^ ^ ^^(l + a;)2 |2 a^ /'"(a;) . (l + a;)" [3 13. li y= f(x), show that ^y-dy^^, diyj^xy d^yjAxf .. ^ da; dx^ [2 ^da;» [3 ' EXPANSION OF FUNCTIONS 101 88. In the preceding derivations of Taylor's and Maclaarin's Theorems, the possibility of the expansion in the proposed form has been assumed. In the remainder of this chapter we shall show how Taylor's Theorem may be derived without such assumption. 89. Rolle's Theorem. If a given function <^(x) is zero when x = a and when x=b, and is continuous between those values, as well as its derivative ^' (a;) ; then 4>'ix) must be zero for some value of X between a and b. Let the function be represented by the curve y = {x). Let OA=a,OB = b. Then accord- ing to the hypothesis, y = when X = a, and when x = b. Since the curve is continuous between A and B, there must be some point P between them, where the tangent is parallel to OX, and consequently ^'(x) = 0. 90. Mean Value Theorem. If/(a:) is continuous from x = a to x = b, there must be some value Xi of x^ for which b — a This may be stated geometrically thus : The difference of the ordinates of two points of a continuous curve, divided by the corresponding differ- ence of abscissas of these points, equals the slope of the curve at some inter- mediate point. In the figure let the curve PBQ represent y=f(x). 102 DIFFERENTIAL CALCULUS l.et 0A= a, OB = b. Then /(6)-/(a) ^M^tanQPif. b — a PM \ At some point of the curve, as R, between P and Q, a tangent can be drawn parallel to PQ. Call 0K= x^. Then the slope of the tangent at R is f'{x^, which equals tan QPM. Hence &) -f ("' )=./ (x,), where a <«!<&. . (1) If we let AB = b — a = h, b = a + h, (1) may be written f(a + h)=f(a)+hf(a + h), where 0<<^<1. . (2) 91. Another Proof. The following method of deriving (2), Art. 84, is important, in that it may be extended to higher derivations of f(x), as appears in Arts. 92, 93. Let R be defined by f(a + h)-f(a)-hR = 0. ... (1) That is, let R denote /(a + h) ~f(a) ' h Consider a function of x whose expression- is the same as (1) with X substituted for h. Call this function ^(rs). That is, ^{x)=f{(i + x)-f{a)—xR (2) Differentiating, <^'(a;) =/'(« + x) — R (3) It is evident from (2) that <^(a;) = 0, when x = h, by (1) ; also {x) = 0, when x = 0. Hence by Eolle's Theorem, Art. 89, <^'(a!) = 0, for some value of x between and h. Calling this value of x, 6h, we have from (3) /'(a + 6h)-R = 0. Substituting this value of R in (1), f{a + h)=f{a) + hf'(a + 6K). EXPANSION OF FUNCTIONS 103 92. Extension of Mean Value Theorem. We may extend the method of the preceding article so as to include the second derivative, and obtain /(a + A) =/ (a) + V («) +?/" (« + Oh). It Define i? by f(a + h)-f{a)-hf'{a)~R = (1) Let ,^{x)=f{a + x)-f{a)-xf{a)-'^n (2) Hence '(x)=f{a + x)—f'(a) — xR, ,i>"(x)=r{a + x)~B (3) From (2) it is evident that 4>(x) = 0, when x = h, by (1); also ^(a;)=0, when a = 0. Hence by Eolle's Theorem, Art. 89, <^'(a-') = 0, for some value, x^, of X, between and h. Also ^' (a;) = 0, when a; = 0. Hence <^" (x) = 0, for some value, X2, between and Xj, that is, be- tween and h. Writing x^ = Oh, we have from (3), f"(a + 0Ji)-R = O. Substituting this value of R in (1), we have f(a + h) =f(a) + hf (a) + |/"(a + Oh). It is to be noticed that it is assumed that/ (x), /'(»), and /"(a;) are continuous from a; = a to a; = a + ^. 93. Taylor's Theorem. This may now be derived by extending the preceding method so as to include the mth derivative. It is assumed that f{x) and its first n derivatives are continuous from x=a to x—a + li. Define R by f(a + h)-f{a)-hf{a) -|/"(a) g/-(a) -|i?=0. (1) 104 DIFFERENTIAL CALCULUS Let ^(a;)=/(a+a;)-/(a)-x/'(a)-|/"(a) ^f'-\a)-^R, af- rix) =/' (a + x)-f' (a) - xf" (a) -^ /»-' (a) - f— r iJ, .^"(o;) =/" (a + x)- f" (a) ^ /"-^ (a) - ,-^i2, <#."-' (a;) =/"-^ (a + x) -/»-• (a) - a;i2, <^" (a;) =/"(« + a;) -i? (2) As in the preceding articles, it is evident that <^(a;) = 0, when x = h, and also when x=0. Hence '{x) = 0, when x = Xj, where < a;i < ^. But '{x) = 0, when cc =0 ; hence This is a definite quantity, unless <^(a) or ij/{a) is zero or infinity. When this is the case, we may, by regarding the fraction as a continuous variable, define its value when x equals a, as its limit when X approaches a. That is, the value of ^^ ' , when a; = a, is defined to be Lim^_, "^ , or what is the same thing, >p{x) There is no difficulty in determining this limit immediately, when the numerator only, or the denominator only, is zero or infinity ; or when one is zero and the other infinity. We will now consider the cases where, for some assigned value of X, the numerator and denominator are both zero or both infinity. The fraction is then said to be indeterminate. 96. Evaluation of the Indeterminate Form - • Frequently a trans- formation of the given fraction will determine its value. rm, x^ + X—2 Q ^ ^ Thus, — I; — - — = - , when x = l. x' — l But if we reduce the fraction to its lowest terms, we have T. a?+ x-2 r- x + 2 3 Lim„i — - — - — = Lim_, — — = - ■ 106 INDETERMINATE FORMS 107 Again, , '^ ^ — = -, when x = 2. Va;-1-1 By rationalizing the denominator, Vx — 1 — 1 » — 2 = Lim„2(Va;— 1 + 1)=2. As another illustration. «°«2e , = 0^when^: cos (9 — sine 4 ^ , ^. cos 2 5 cos^0— sin^e But Lim„_!r 7. : — 2 ~ Lini„_i. ;:; : — x ''-4COS — sm 6 ^-4 cos^ — smS = Limj^i(cos5 + sin 6)= cos- + sin-= ^^_ 4 4 4' The Differential Calculus furnishes the following general method: 97. Form a new fraction, taking the derivative of the given numerator for a new numerator, and of the given denominator for a new denomi- nator. The value of this new fraction, for the assigned value of the variable, is the limiting value of the given fraction. We will now show how this rule is derived. (h(x) Suppose the fraction ^^ ' = -, when x = a; that is, {a) =0, and By Art. 95 the required value of the fraction is the limit of (a) + h'{a + Oh), ^{a + h) =ip(a) + hxfi'{a + OJi), where 6 and 61 are proper fractions. 108 DIFFERENTIAL CALCULUS But since ^(a) = and ip(a) = 0, we have (a + h) ^ h^'{a + Bh) ^ ^'{a + Oh) _ Hence Lim^<^(« + ^0 = ^>), which is the theorem expressed by the rule. If <^'(a) = and i/''(a) = 0, it follows likewise that V'(a + /i) «//"(«) that is, the process expressed by the rule must be repeated, and as often as may be necessary to obtain a result which is not indeter- minate. For example, let us find the limiting value of the fraction in Art. 96. (a;) log a; <» t, n • -s-^ = ° = — , when a; = 0. i/f(a;) cot X oo 1 dK^ ^' dx\dxj d3?' and (1) becomes _^ < 0. Hence when ^ = 0, and ^ < 0, (2) dx dar there is a maximum value of y. By similar reasoning we have the case (6), when -r^ > 0. dop Hence when ^ = 0, and ^ > 0, (3) dx aar there is a minimum value of y. For example, let us find the maximum and minimum values of the function ^ 3 Put y = ^-2a? + 3x + l. Then ^ = a^-4a! + 3, (4) dx ^ = 2x-4 (5) dor Putting (4) = 0, a^-4a; + 3 = 0, whence x = l or 3. Substituting those values of x in (5), we find whena;=l, g=-2<0; whena; = 3, |^ = 2>0. 116 DIFFERENTIAL CALCULUS Hence by (2) and (3), when x = l, when a; = 3, y has a maximum value ; y has a minimum value. From the given function we find that the maximum value of y is y = 2^, and the minimum value of y, y = 1. 103. In exceptional cases it may happen that a value of x given by ^ = 0, makes ^ = 0, so that neither (2) nor (3), Art. 102, is satisfied. dx dx- This would be the case for a point of inflection E. (see Art. 158) whose tangent is parallel to OX. Here the ordinate RL is neither a maximum nor a minimum. But there may be a maximum or minimum value of y, even when ^=0. This is more fully dx^ considered in Art. 106. The following article is also appli- cable to such cases. 104. Second Method of determining Maxima and Minima. Maxim::, and minima may be determined from the first derivative — alone, d^v ^^ without using — ^. daf We have seen in Art. 102 that when ?/ is a maximum, as at P, the slope, that is, -^ , changes from + to — ; and when y is& mini- mum, as at Q, rr changes from — to + . (It is understood that we dx pass along the curve from left to right.) MAXIMA AND MINIMA OF FUNCTIONS 117 By examining the form of -^, which should be expressed in factor dx form, we may determine whether it changes from + to — , or from — to + , for any assigned value of a;. Let us apply this method to the example in Art. 102, ^ = a;2 - 4a; + 3 = (a; - l)(a; - 3). dx Here — can change sign only when a; = 1 or a; = 3. dx & & J By supposing x to change from a value slightly less, to one slightly greater than 1, we find that (x — 1) changes from — to + ; but since the factor (x — 3) is then negative, it follows that -^ changes from dx + to — , when a; = 1, and denotes a maximum. In the same way, we find that -^ changes from — to +, when a; = 3, and denotes a mini- da; mum. Again, consider the function y = (x — 4)°(a; + 2)*. Differentiating and writing the result in factor form, ^ = 3 (3 a; - 2)(a; - 4)^(a; + 2)1 When a; = -, -^ changes from — to + . 3 dx When a; = — 2, -^ changes from + to — • dx When a; = 4, -^ does not change sign, dx dy dx since (x — 4)* cannot be negative. 2 Hence we conclude that y is a minimum when a; = -; a maximum o when a; = — 2 ; but neither a maximum nor minimum when a; :?= 4. As this method does not require -^,, it is preferable to that dar of Art. 102, when the second differentiation of y involves much work. 118 DIFFERENTIAL CALCULUS EXAMPLES 1. Find the maximum value of 32x — a^. Ans. 48. Find the maximum and minimum values of the following functions. 2. 2oi? — Za? — 12x-\-12. Ans. a; = — 1, gives a maximum 19. X = 2, gives a minimum — 8. 3. 23? — lla?-\-12x-'rlO. Ans. x = -, gives a maximum 13|^, 2 3' a; = 3, gives a minimum 1. 4. a;^ + 9 (a — x)^. Ans. x=—, gives a maximum — . 3a . . . 9a= x= — , gives a minimum — -— . 4 ^ 16 5. (a!-l)(a!-2)(a;-3). Ans. X = 2 , gives a maximum V3 3V3- 1 2 x = 2-\ -, gives a minimum — V3 3V3 6. 2 (3 a; + 2)^ — 3 x*. Ans. x = 2, gives a maximum 80. X* — 4:0^ + 8x 8 7. Show that -~- has no maximum nor minimum. X— 1 8. ^ + _^,wherea>6. Ans. X = -, gives a maximum i^^JlJLL a + b a • x = -, gives a minimum ^'^ ~ ) a — a ' 9. Show that the greatest value of _2S^ is — . a^ ne 10. Show that the greatest value of cos 2 ^ + sin is — . 8 MAXIMA AND MINIMA OF FUNCTIONS 119 11. Show that the maximam and minimum values of sin!' d + sin' f- - e] are - and -. \3 J 2 2 12. Find the maximum value of a sin x + b cos x. Ans. ■\/aF+W. 13. Find the maximum value of tan~' x — tan"^ -, the angles being taken in the first quadrant. a fan-' ? 4' 14. Show that the least value of a? tan- B-^-h' cot^ 6 is the same as that of aV' + ft^e""*, and equal to 2 ah. 15. ti = ^'^~' . Ans. A minimum when a; = - . ^ a-2x 4 16. y = {x-l)\x + 2f. Ans. A maximum when a; = — - ; a minimum when x = l; 7 neither when x = — 2, 17. y=.{x-2)\2x + l)\ Ans. A maximum when a; = — - ; a minimum when x = --, 2 lo neither when x — 2. 105. Case where ^ = oo . It is to be noticed that -^ may change . dx dx sign by passing through infinity instead of zero. Hence if -^=oo, dx for a finite value of x, this value should be examined, as well as those given by ^ = 0. (Za; 120 DIFFERENTIAL CALCULUS For example, suppose 2 Then dy_ dx 2 b 2,{x- -cy hence we have Y ? = -' dx when x = c. It is evident that when x=c, dy dx changes from + to — , indicating a maximum value of y, which is a. The figure shows the maximum ordinate PM, corresponding to a -^ cusp at p. On the other hand, suppose y = a— b{x — c)^. Then dy^ dx 3{x-c)^ : 00 when i dy \ But as -^ does not change sign when x = c, there is no maxi- dx mum nor minimum. The corresponding curve is shown in the figure. Y MAXIMA ANU MINIMA OF FUNCTIONS 121 EXAMPLES Find the maximum and minimum values of the two following functions : 1. y = {x + l)^x-5y. A71S. A minimum when a; = 5; a maximum when a; = ; a minimum when x = — l. 2. y = {2x-a)^{x-a)i- 2 a Ans. A maximum when x = — ; a minimum when x = a. o 106. Conditions for Maxima and Minima by Taylor's Theorem. Suppose the function /(x) to be a maximum when x = a. Then, by the definition in Art. 101, f{a)>f(a + h), and also /(a) >f(a — Ji), where h is any small but finite quantity. Now, by the substitution of a for x in Taylor's Theorem, we have f(a + h)-f(a)= A/(a) + |/"(a) + |/-(a) + .... . (1)* /(a-70-/(a) = -7i/'(a) + |/"(a)-|/"'(a) + -. . (2) By the hypothesis /(a + /i) —f(a) < 0, and also f{a — li)—f{a)<0. Hence the second members of both (1) and (2) must bg negative. * The rigorous form of Art. 93 may be used here without auy change in the context. 122 DIFFERENTIAL CALCULUS By taking 7i sufficiently small, the first term can be made numeri- cally greater than the sum of all the others, involving 7i^, /i', etc. Thus the sign of the entire second member will be that of the first term. As these have different signs in (1) and (2), the second mem- bers cannot both be negative unless /'(a) = 0. Equations (1) and (2) then become /(a + A) -/(a) = I /"(«) + |V"'(a) + • • •, /(a-A)-/(a) = |/'(a)-||/'"(a)4-.... The term containing W now determines the sign of the second members. That these may be negative, we must have /"(a)<0. If then /'(a) = and /"(«)< 0, /(a) is a maximum. Similarly, it may be shown that if /'(a) = and /"(a)>0, /(a) will be a minimum. If /'(a) = and /"(a) = 0, . similar reasoning will show that for a maximum we must also have /"'(a) = and f\a)<0 ; and for a minimum /"'(a)=0 and /»>0. The conditions may be generalized as follows : Suppose that /'(«) = 0, /"(a) = 0, /"'(a)=0, - .f(a) = 0, and that f"^\a) is not zero. Then if v^s even, /(a) is neither a maximum nor a minimum. If n is odd, /(a) will be a maximum or a minimum, according as /''+Xa)<0 or >0. MAXIMA AND MINIMA OF FUNCTIONS 123 PROBLEMS IN MAXIMA AND MINIMA 1. Divide 10 into two such parts that the product of the square of one and the cube of the other may be the greatest possible. Let X and 10 — a; be the parts. Then v? (10 — a;)' is to be a maxi- mum. Letting m = x^ (10 — «)', we find — =5 a;(4 - a;)(10-a;)2 = 0, from which we find that m is a maximum when a; = 4. required parts are 4 and 6. Hence the 2. A square piece of pasteboard whose side is a has a small square cut out at each corner. Find the side of this square that the remainder may form a box of maximum contents. Let X = the side of the small square. Then the contents of the box will be (a — 2 a;)^ x. Representing this by u, we find that m is a maximum when x= -, which is the required answer. 3. Find the greatest right cylinder that can be inscribed in a given right cone. 'LQtAD = a,DC=h. Let x — DQ, the radius of the base of the cylinder, and y = PQ, its altitude. From the similar triangles ADC, PQC, we find y y = l(p-x). The volume of the cylinder is a Tr3?y = Tr-3?{b- ■X). This will be a maximum when u=ba^—a^ is a maximum. This is found to be when x required cylinder. From this, y = -, the altitude of the cylinder. o I 6, the radius of the base of the 124 DIFFERENTIAL CALCULUS 4. Determine the right cylinder of the greatest convex surface that can be inscribed in a given sphere. Let r = OP, the radius of the sphere ; x = OR, the radius of base of cylinder ; and y = PR, one half its altitude. From the right triangle OPR we have The convex surface of the cylinder is 2 Tra; • '^ ' -y- ■■ 4 irX Vj^ — x^. We may put u equal to this expression, and determine the value of x that gives a maximum value of u. But the work may be shortened by the following considerations : 4 ttx s/'t^ — x^ is a maximum, when and when its square X V?"^ — a^ is a maximum ; xy/r^ — x^ is a maximum, 1^0? — X* is a maximum.* Hence we may put u = rV — a;*, from which we find m is a maximum, when a; = V2 From this y = — =, giving for the altitude of the cylinder, V2 Another Metlwd. The equations The convex surface = 4 irxy, u = xy, (1) a? + f=r', (2) may be used without substituting in (1) the value of y from (2). * Since we are only concerned with th.e positive root of Vr^ — x''. MAXIMA AND MINIMA OF FUNCTIONS 125 Differentiating (1), ^=^y + x% (3) dx dx and differentiating (2), x + y^^O, ^=_^. dx dx y Substituting in (3), ^ = y - ^ = t^' . dx y y Since x and y are positive quantities, it is evident that when x=y, — changes from + to — , giving a maximum value of u. Combining a; = 2/, with (2), we have X = — , y = — , as before. V2 V2 In some problems this method has some advantages over the first. 6. Divide 48 into two parts, such that the sum of the square of one and the cube of the other may be a minimum. Ans. 42|, 5^. 6. Divide 20 into two parts, such that the sum of four times the reciprocal of one and nine times the reciprocal of the other may be a minimum. Ans. 8, 12. 7. A rectangular sheet of tin 15 inches long and 8 inches wide has a square cut out at each corner. Find the side of this square so that the remainder may form a box of maximum contents. Ans. If in. 8. How far from the wall of a house must a man, whose eye is 5 feet from the ground, stand, so that a window 5 feet high, whose sill is 9 feet from the ground, may subtend the greatest angle ? Ans. 6 ft. 9. A wall 27 feet high is 8 feet from the side of a house. What is the length of the shortest ladder from the ground over the wall to the house? Ans. 13Vl3=46.87 ft. 126 DIFFERENTIAL CALCULUS 10. A person being in a boat 5 miles from the nearest point of the beach, wishes to reach in the shortest time a place 5 miles from that point along the shore ; supposing he can run 6 miles an hour, but row only at the rate of 4 miles an hour, required the place he must land. Ans. 929.1 yards from the place to be reached. 11. Find the maximum rectangle that can be inscribed in an ellipse whose semiaxes are a and 6. Ans. The sides are aV2 and 6V2; the area, 2ab. 12. A rectangular box, open at the top, with a square base, is to be constructed to contain 500 cubic inches; What must be its dimensions to require the least material ? AnS. Altitude, 5 in ; side of base, 10 in. 13. A cylindrical tin tomato can is to be made which shall have a given capacity. Find what should be the ratio of the height to the diameter of the base that the smallest amount of tin shall be required. Ans. Height = diameter. 14. What are the most economical proportions for an open cylin- drical water tank, if the cost of the sides per square foot is two thirds the cost of the bottom per square foot ? Ans. Height = f diameter. 15. (a) Find the altitude of the rectangle of greatest area that can be inscribed in a circle whose radius is r. Ans. rA/2 ; a square. (6) Find the altitude of the right cylinder of greatest volume that 2?- can be inscribed in a sphere whose radius is r. Ans. V3 16. (a) Find the altitude of the isosceles triangle of greatest area 3,. inscribed in a circle of radius r. Ans. -— ; equilateral triangle. (6) Find the altitude of the right cone of greatest volume inscribed An- in a sphere of radius r. Ans, ~ . o MAXIMA AND MINIMA OF FUNCTIONS 127 17. (a) Find the altitude of the isosceles triangle of least area circumscribed about a circle of radius r. Ans. 3 r ; equilateral triangle. (b) Find the altitude of the right cone of least volume circum- scribed about a sphere of radius r. Ans. 4 r. 18. A right cone of maximum volume is inscribed in a given right cone, the vertex of one being at the center of the base of the other. Show that the altitude of the inscribed cone is one third the altitude of the other. 19. Find the point of the line, 2x + y = lQ, such that the sum of the squares of its distances from (4, 5) and (6, —3) may be a minimum. Ans. (7, 2). 20. Find the perpendicular distance from the origin to the line - + 7 = 1, by finding the minimum distance. Ans. ,— 21. A vessel is sailing due north 10 miles per hour. Another vessel 190 miles north of the first is sailing 15 miles per hour on a course East 30° South. When will they be nearest together, and what is their least distance apart ? Ans. In 7 hrs. Distance 15V67 = 113.25 mi. 22. A vessel is anchored 3 miles ofE the shore. Opposite a point 5 miles farther along the shore, another vessel is anchored 9 miles from the shore. A boat from the first vessel is to land a passenger on the shore and then proceed to the other vessel. What is the shortest course of the boat ? Ans. 13 mi. 23. The velocity of waves of length A. in deep water is propor- tional to \/ - + - , where a is a certain linear magnitude. Show that ^ a \ the velocity is a minimum when A = a. 24. Assuming that the current in a voltaic cell is C= , E r+ R being the electromotive force, r the internal, and R the external, resistance; and that the power given out is P= RC; show that P is a maximum when R = r. 128 DIFFERENTIAL CALCULUS 25. From a given circular sheet of metal, to cut out a sector, so that the remainder may form a conical vessel of maximum capacity. Ans. Angle of sector = f 1 _ "^^ |2 u = 66° 14'. \ 37 26. Find the height of a light on a wall so as best to illuminate a point on the floor a feet from the wall ; assuming that the illumi- nation is inversely as the square of the distance from the light, and directly as the sine of the inclination of the rays to the floor. Ans. V2 27. At what point on the line joining the centres of two spheres must a light be placed, to illuminate the largest amount of spherical surface ? Ans. The centres being A, B ; the radii, a, h ; and P the required point; AP':PB^=a'':h\ 28. (a) The strength of a rectangular beam varies as the breadth and the square of the depth. Find the dimensions of the strongest beam that can be cut from a cylindrical log whose diameter is 2 a. Ans. Breadth = — • Depth = 2 a^ /I. V3 \3 (6) The stiffness of a rectangular beam varies as the breadth and the cube of the depth. Find the dimensions of the stiffest beam that can be cut from the log. Ans. Breadth = a. Depth = a V3. 29. The work of propelling a steamer through the water varies as the cube of her speed. Find the most economical speed against a current running 4 miles per hour. Ans. 6 mi. per hr. 30. The cost of fuel consumed in propelling a steamer through the water varies as the cube of her speed, and is $ 25 per hour when the speed is 10 miles per hour. The other expenses are $100 per hour. Find the most economical speed. Ans. -v/2000 = 12.6mi. per hr. 31. A weight of 1000 lbs. hanging 2 feet from one end of a lever is to be raised by an upward force at the other end. Supposing the lever to weigh 10 lbs. per foot, find its length that the force may be a minimum. Ans. 20 ft. MAXIMA AND MINIMA OF FUNCTIONS 129 32. (a) The lower corner of a leaf, -whose width is a, is folded over so as just to reach the inner edge of the page. Find the width of the part folded over, when the length of the crease is a minimum. Ans. I a. (6) In the preceding example, find when the area of the triangle folded over is a minimum. Ans. When the width folded is | a. 33. A steel girder 25 feet long is moved on rollers along a pas- sageway 12.8 feet wide, and into a corridor at right angles to the passageway. Neglecting the horizontal width of the girder, how wide must the corridor be, in order that the girder may go around the corner ? Ans. 5.4 ft. 34. Find the altitude of the least isosceles triangle that can he circumscribed about an ellipse whose semiaxes are a and b, the base of the triangle being parallel to the major axis. Ans. 3 b. 35. A tangent is drawn to the ellipse whose semiaxes are o and b, such that the part intercepted by the axes is a minimum. Show that its length is a + 6. CHAPTER XI PARTIAL DIFFERENTIATION 107. Functions of Two or More Independent Variables. In the pre- ceding chapters differentiation has been applied only to functions of one independent variable. We shall now consider functions of more than one variable. Let _ u=f(x,y) be a function of the two independent variables x and y. Since x and y are independent of each other, we may suppose x to vary while y remains constant, or y to vary while x remains con- stant ; or we may suppose x and y to vary simultaneously. We must distinguish between the changes in u resulting from these dif- ferent suppositions. Let A^u denote the increment in u resulting from a change in x only, and A^u the increment in u from a change in y only. Let Am, called the total increment of u, be the in- , N p' crement when x and y both change. Suppose u the area of a rectangle whose sides are x and y. Then u = xy. Ay AjW A,u Ax If X changes from OA to OA', while y remains con- stant, u is increased by the rectangle AM. That is, Ajj, = area AM. 130 PAKTIAL DIFFERENTIATION 131 If y changes from OB to OB', while x remains constant, u is increased by the rectangle BN. That is, AjW = area BN. If X and 2/ both change together, we have for the total increment of u, Am = area AM+ area BN+ area MN. 108. Partial Differentiation. This supposes only one of the inde- pendent variables to vary at the same time, so that the differentia- tion is performed by the same rules that have been applied to functions of a single variable. If we differentiate u=f(x, y), supposing x to vary, y remaining constant, we obtain dx If we differentiate, supposing y to vary, x remaining constant, we 1 . . du obtain — dy • • du du The derivatives, — , — .thus deiived, aj:e oalleA partial deriva- dx dy' fives, and a special notation, — , — , is used for them. ' ^ ' dx' dy' For example, if u = a^ + 2 x^y — j/', -- = 3 a^ + 4 a;?/, the avderivative of u. dx — = 2a? — S y^, the 2/-derivative of u. dy In general, whatever the number of independent variables, the partial derivatives are obtained by supposing only one to vary at a time. EXAMPLES Derive by partial differentiation the following results : , XV du , du 1. u = — 2_, x— + y — = u. X + y ox ay 2. z = (ax' + 2 bxy + cy^, (hx + cy)j- = (ax + by) j-. 132 DIFFERENTIAL CALCULUS 3. u = {y-z)(z-x){x-y), ^ + J5+^* = 0. dx dy dz . , dr dr ^^ + y > '^ dx dy 5. u = \o^{a? + ax^y ->rhxy^ + cf), x— + y — = 3. 6. M = ■^^ h-, a'5- + 2/T" +^I" = "• z a; 2/ ax ay az a; + 2/ ' y dx dy x' + y' 9. z ={x +y){x' — 2/^", 2// + « 5^ = «• 5a; 5?/ m . ^_,. (|)-+(?-"r+ (^ + 2/^ + ^7 11. « =log„a; + log^2/> a; log a; ^ + 2/ log 2/7^ = 0. 9a; dy 12. M= e' sin 2/ + e" sin a;, ^Y+ ri^Y^ e^ + e-" + 2 e'+" sin (a; + 2/). 13. M = log (tan X + tan 2/ + tan 2), sin 2a; ^ + sin 22/ ^ + sin 22 ^ = 2. oa; dy dz PARTIAL DIFFERENTIATION 133 109. Geometrical Illustration of Partial Derivatives. Let z =f(x, y) be the equation of the surface APGH. The ordinate PN is thus given for every point N in the plane XY. Let APB and CPD be sec- tions of the surface by planes through P, parallel to XZ and YZ, respectively. If X and y both vary, P moves to some other position on the surface. If X vary, y remaining con- stant, P moves on the curve of intersection APB. dz Hence — is the slope of the curve APB at P. dx If y vary, x remaining constant, P moves on the curve CPD. Hence — is the slope of the curve CPD at P. 110. Equation of Tangent Plane. Angles with Coordinate Planes. In the figure of the preceding article, let P be the point {x', y', z') ; PT, the tangent to APB in the plane APNM; and PT, the tangent to CPD in the plane CPNL. It is evident from the preceding article that the equations of PT are and of PT', z-z' = ^£Xx-x'), y==y', dz' z-z' = — (y-y'), x = x'. oy' (1)* (2) ^, ^ denote the values of ^, — , reapectlvely, for (»', !/', z')- dx'' dy dx dy 134 DIFFERENTIAL CALCULUS The plane tangent to the surface at P contains the tangent lines PT and PT'^ Its equation is ^^(^•'^')4f(M-'f'i^k^-^'=|;(.-.')+'^(2/-y) (3) For (3) is of the first degree with respect to the current variables X, y, z, and is satisfied by (1), and also by (2). Tlie equations of the normal through P are those of a line through (x', y', z') perpendicular to (3). Its equations are i^- !lil!-^' «^-x' y-y'_ , ,x m ^^' ''r '^' to' 3? The angles made by the tangent plane ivith the coordinate planes are equal to the inclinations of the normal to the coordinate axes. By analytic geometry of three dimensions, the direction cosines of the line perpendicular to (3) are proportional to ^ ^ _1 dx'' dy' Hence, if «, p, y, are the inclinations of the normal to OX, Y, OZ, respectively, cos a _ cos p _ cos y ~dF~^z[ "HT dx dy' (5) Also cos^ a + cos^ /8 + cos^ y = 1 (6) Prom (5) and (6) we find, dropping the accents, dx cos'' « = [dxj yey^ PARTIAL DIFFERENTIATION 135 cos^ j8 = \dy) ■For the inclination of the tangent plane to XT, we have from (7), -v-Hs)"-(l)" '«) and tan^ y=( — i +1 — ^ \dxj ^\dy The term sl(^e used in geometry of two dimensions may thus be extended to three dimensions, as the tangent of the angle made by the tangent plane with the plane XY. In this sense, — p»=V(D'HIJ EXAM PLES 1. Find the equations of the tangent plane and normal, to the sphere a:F + y^ + z- = d\ at (a;', y,' z'). Hence Substituting in (3), z — z'—— — (x—x') —^(y — y'), dz _ X dx z' dz _ y_ dy z dz' _ x' dx' z'' dz' y' dy' z' _,'-_-', ^-K_V^_^ xx' + yy' + zz' = x'^ + y"'-^-z" = a\ Ans. 136 DIFFERENTIAL CALCULUS From (4) we find for the normal (x - x') ^ = (y-y')^- = z- z', X y ^-1 = ^-1 = ^-1, ^ = ^ = ?-. Ans. x' y z' x' y' z 2. Find the equations of tangent plane and normal to the cone, 3 a;2 _ y2 + 2 ^2 = 0, at (a;', y', z'). Ans. 3xx' — yy'+2zz' = 0, ^^^^ = ^^i:^ = ?j=ll . 3x' — 2/ 2 z' 3. Find the equation of the tangent plane to the elliptic parabo- loid, z=3x' + 2y% at the point (1, 2, 11). Ans. 6x+8y — z = ll. 4. Find the equations of tangent plane and normal to the ellipsoid, a^ + 2y'' + 3z' = 20, at the point a; = 3, y = 2, z being positive. Ans. 3x + iy + 3z = 20; x = z + 2, 3y = Az + 2. Find the slope of this tangent plane. A7is. f . 5. Find the equation of the tangent plane to the sphere, x^ + y^ + z'-2x + 2y = l, at (a;', y', z'). Ans. xx' + yy' + zz'—x — x' +y + y' = 1. 111. Partial Derivatives of Higher Orders. By successive differ- entiation, the independent variables varying only one at a time, we may obtain d^u d^u d'u d*u ■ d^' 'df' a^' ly"''"' If we differentiate u with respect to x, then this result with respect to y, we obtain —(■^\ which is written ^ ^ . "' dy\dxj dydx PARTIAL DIFFERENTIATION 137 dhi Similarly, is the result of three successive differentiations, dydar two with respect to x, and one with respect to y. It will now be shown that this result is independent of the order of these differentiations. In other words, the operations — and — are commutative, dx dy That is ^^** ^^^ ^'" ^°" ^^^ dydx dxdy dydx? dxdydx dx'dy 112. Given u=f{x,y), (1) to prove that If^UA/^M. dy\dxj dx\dyj Supposing x to change in (1), y being constant, Am ^ f(x + Aa;, y) —f(x, y) ^ Ax Ax (2) Now supposing y to change in (2), x being constant, A/'^^ = /fa + Ax,y + Ay) —f(x, y + Ay) —f{x + Ao;, y) +f{x, y) ^ Ay\AxJ Ay Ax Reversing the above order, we find An ^ f{x, y + Ay) -f{x, y) ^^^^ Ay Ay _A/Au\ f(x + Ax,y + Ay) —f(x + Ax, y) — /(x, y + Ay)+ f(x, y) Ax\Ay) Ax Ay Hence AA^VAf^) (3) Ay\AxJ Ax\AyJ ^ ' The mean value theorem, (2), Art. 90, may be expressed in the form — = fix-^e-Ax), where m =/(a;). 0<^<1. Ax 138 DIFFERENTIAL CALCULUS In the present case, where u =f(x, y), ^=Ux + e,.^x,y).* Ax Similarly, -^ =/„(«, y + Oa-Ay), Ay and ixKA) ^ -^'"^^ "*" ^^" ^^' ^ "^ ^'' ^^'^ " By (3) f^{x+0,-Ax, y + e,-Ai/) =/,/x + e4-Aa;, y + e,-Ay). Taking the limits as Ax, Ay, approach zero, and assuming the functions involved to be continuous, ny, , ■ d /5m\ 9 /SmX d-u 8^U That IS, — — )= — ( — , or = . dy\dxj dx\dyj dydx dxdy This principle, that the order of differentiation is immaterial, may be extended to any number pi differentiations. Thus, &>u ^_d^fdu\ ^ d' f8u\ ^ _3^ dyda? dydx\dxj dxdy\dxj dxdydx dx\dydxj dx\dxdyj dx'dy It is evident that the same is true of functions of three or more variables. */.(*, 2/) = ^/(a:,^), fv(x,y)=^f(x, y), dx ay PARTIAL DIFFERENTIATION 139 EXAMPLES Verify -^ = -^ in Exs. 1-3. ax ay dy dx 1. u = ^^-±^, 2. u = xy\og-. 3. u = (x + y)e'-''. ay + bx y \ ^yj Derive the following results : 9^M d^u d*u 4. M = ax* + 6 hii?y'' + ct/*, dx'dy^ dxdydxdy dxdy'dx ' 5. «=iog(^+,o, S+|:=«- 6. «=(3a; + 2/V+sin(2a;-2/), find — , — 6—. 9a;^ dx dy dy' 7. « = ?'+log?, find x^^+2^-^ + 2/^^ 2/ 3/ 9a; dxdy dy' 8. 2=x2t.a,n-i^_2/2tan-i-. _i^ = ^!zil'. x y ' dxdy or + y' 9. g = (r'.+r-»)cosne, ^ + 1^+1 g = 0. 9r r or r^ 90'' 10. tt = log (e" +e"+e'), __d^u_ ^ gg'+i'+^-S". * ^ ^' 9a;93/9« ? ? ? fl6„ ! ? ; 13. M = y Ve^ + « W + a;^2/ V, ^" =e^ + e' + e^ 14. M = sin (2/ + «) sin (z + x) sin (a; + y), 9«w dafdy'dfi^ = 2cos(2a; + 22/ + 2»). 9a; 9t/ 9« 140 DIFFERENTIAL CALCULUS 113. Total Derivative. Total Differential. In Art. 107 we have referred to the change in. u when x and y vary simultaneously. This change is called the total increment of u. Thus the total incre- mentof u=f{x,y) is Au=f(x + Aa;, y + Ay) - f(x, y). The terms total derivative and total differential are also used. For example, let u = a?y — 3x'y\ (1) and suppose x and y to be functions of a variable t. Differentiating with respect to t, = a?'M + 3a?y ^-6a^y'^-6xy' — dt dt dt dt = (5x'y-&xy')^^+(x'-&^yfjL (2) But from (1) we find nfti fit/ — = 3 K^u — 6 xy% -— = x^ — &y?y. ox ay So that (2) may be written du _ d,u dx 8u dy ,„% dt~dxdi dydt ^ ^ If we had used differentials in differentiating (1) we should have obtained du = — dx + -—dy . . (4) dx dy " ^ ^ — in (2) and (3) is called the total derivative, and du in (4) the total dt differential, of u. We proceed to show that (3) and (4) are true for any function of x and y. PARTIAL DIFFERENTIATION 141 Noticing that A?t is the total increment of u, and AjM, \u, the partial increments, when x and y vary separately, u =/(», y), X and y being functions of t. u'=f{x + \x,y), u"=f(x + Ax,y + \y). Then A^m = m' — m, \u' = u" — u', Au = u" — u. Hence Am = Aji + /\,,u', and Am ^ A^ Ax Ay Ay At Ax At Ay At' Taking the limits of each member, as At, and consequently Ax, Ay, approach zero, du^dudx dudy rg. dt dx dt dy dt ' since the limit of u' is u. This may be written in the differential form du = ^dx + pdy (6) dx ay In the same way, if u =/(», y, z), where x, y, % are functions of t, we find dM^Budx ^dy hudz _ ,r^. dt dx dt dy dt dz dt' and au = ^^dx+^^dy + ^^dz (8) dx ay oz We may write in (8) |^da; = d.M, ^dy = d,u, p dz = d^, dx dy oz giving du = dji + d^u + d^u, that is, the total differential of u is the sum of its partial differentials. 142 DIFFERENTIAL CALCULUS This principle, as expressed by du — dji + d^u, may be illustrated by the figure of Art. 107, from which we have Am = \^u + ^.yU + area MN, that is, Aw = A^M + A„M + Aa; Ay. As Aa; and A?/ approach zero, the last term diminishes more rap- idly than the others, and we may write Am = AjM + AjM, approximately, the closeness of the approximation increasing as Aa; and Ay approach zero. If in (5) we suppose t = x, then u=f{x, y), y being a function of x ; and (5) becomes dM^du^dud^ , dx dx dy dx Similarly, if in (7), t = x, u=f(x,y,z), y and z being functions of x ; whence du ^ du _^ du dy _^ du dz dx dx dy dx dz dx ^ EXAMPLES Find the total derivative of u by (5) or (7) in the three following : 1. u=f{x,y,z), where x = f, y = f, «=-. dt " dx dy e~dz' 2. M = log (x'~ y^), where x = a cos t, y = a sin t. ^ = - 2 tan 2 (. dt 3. M = tan-' -, where x = 2t, y = l-f. ^^ dt 1 + f. PARTIAL DIFFERENTIATION 143 Apply (10) to the two following : 4. u =f(x, y, z), where y=^~x, z = a? — oi?. da; dx ^ dy ^ ' dz du 5. M = tan-i^, where y = 3-x', z = l-3x', — = dx 1 + a;^' Find the total differential by (6) or (8) in the following : 6. u = ax^ + 2bxy + cy% du = 2{ax + by)dx + 2{bx + cy)dy. 7. ^* = a;>°s^ du = uf^-^lldx + ^-^SJo,^y \ X y _ , sin A (a; + y) , sin y dx— sin x dy 8. M = log^ — |-) % du = i-- sin ^(x — y) cos x — cos y 9. u = aoe' + hij- + cz^ + 2fyz + 2gzx + 2hxy, du=2 (ax + hy+gz) dx + 2(hx + hy+fz) dy+2{gx +fy + cz)dz. 10. u = x"", du = x'"'~^ {yz dx + zx log xdy + ooy log x dz) . 11. M = tan2a;tan22/tan22, au = 4.u(~-^^+ -^^ + -J^^. \sin 2 a; sva.2y sin 2 zj If the variable t in (5) and (7) denotes the time, we have the re- lation between the rates of increase of the variables. For illustration consider the following example : 12. One side of a plane triangle is 8 feet long, and increasing 4 inches per second; another side is 6 feet, and decreasing 2 inches per second. The included angle is 60°, and increasing 2° per second. At what rate is the area of the triangle increasing ? 144 DIFFERENTIAL CALCULUS The area A = -6c sin A, from whieli dA c • .dh , h . ,dc , he ,dA — = - sin^ f- - sin^ — — cos^ — dt 2 dt 2 dt 2 dt = lsinA-\ + lsmA.-\ + ^cosA^ 2 3 2 6 2 90 = .4934 sq. ft. = 70.05 sq. in. per sec. 13. One side of a rectangle is 10 inches long, and increasing uni- formly 2 inches per second. The other side is 16 inches long, and decreasing uniformly 1 inch per second. At what rate is the area increasing ? Ans. 20 sq. in. per sec. At what rate after the lapse of 2 seconds ? Ans. 12 sq. in. per sec. 14. The altitude of a circular cone is 100 inches, and decreasing 10 inches per second, and the radius of the base is 50 inches and increasing 5 inches per second. At what rate is the volume in- creasing ? Ans. 15.15 cu. ft. per sec. 15. In Ex. 12, at what rate is the side opposite the given angle increasing ? Ans. 8.63 in. per sec. 114. Differentiation of an Implicit Function. (See Art. 66.) The derivative of an implicit function may be expressed in terms of partial derivatives. The equation connecting y and x, by transposing all the terms to one member, may be represented by <^(«',2/)=0 (1) Let u = ij>{x, y). From (9), Art. 113, we have for the total derivative of u, du _ du du dy dx dx dy dx PARTIAL DIFFERENTIATION 145 But by (1) X and y must have such values that u may be zero, that is, a constant; and therefore its total derivative— must be zero. dx Hence and bu . dx du dy _ dy dx --0, dy^ du du dx dy ■om 3?y' + 3?f = :a'. (2) For example, find -^ from dx Let u = 3?y' + a^2/^ — a?. dx dy By (-2) dy_ Sa?y^+2xf ^ ^xy + 2f ' dx 23?y+Z3?y' 2x^ + 3xy' In the same way find the first derivatives in the examples of Art. 67. 115. Extension of Taylor's Theorem to Functions of Two Inde- pendent Variables. If we apply Taylor's Theorem to f(x+h,y+k), regarding x as the only variable, we have f(x+ h,y + Tc) =f{x, y + k) + h^f{x, y + k) 146 DIFFERENTIAL CALCULUS Now expanding f{x, y + fc), regarding y as the only variable, f{x, y + k) =f(x, y) + k—f(x, y) + -- -r-J{x, y) + ■■■■ oy \± oy' Substituting this in (1), f(x + h,y + k)= f(x, y)+h— f {x, y) + k~ f(x, y) ax ay H \f^fix,y)+2Uk^^fix,y)+k^f^Jix,y) + •••• (2) This may be expressed in the symbolic form thus : f(x + h,y + k) =f(x, y) + (h£ + k£\ fix, y) \2\ dx dyj-'^ ' ^^ \3\ dx dyy^"^' ' wheref A \-k — | is to be expanded by the Binomial Theorem, as V 3a; dy) if h— and k — were the two terms of the binomial, and the result- 5a; dy ing terms applied separately to /(a;, y). 116. Taylor's Theorem applied to Functions of Any Number of In- dependent Variables. By a method similar to that of the preceding article we shall find f{x + h,y + k,z + V)=f{x,y,z) + (h~ + k^ + l£jf{x,y,z) This expansion may be extended to any number of variables, PARTIAL DIFFERENTIATION 147 EXAMPLES 1. Expand log (a; + h) log (y + k). 'Letu=f(x,y) = logx\ogy, ^=-^^, J^ = _2S^, dx X oy y d-u_ logy d^u _ 1 d^u ^ log a dx^ 3? ' dydx xy' dy' y^ By (2) , Art. 115, log (x + h) log (y + k) = log a; log y , h^ , fc, h^ ■, , hk k^ , . + -log2/ + -loga;- ^— ; log 2/ H — -,loga;+ ••- X y 2x~ xy 2y- 2. (a; + ft)^ (2/ + A;)^ ^a?y'- + 3 hxY + 2 Aix^'y + 3h^xy^ + Qlikx^y + W:i?+--: 3. sin [(cc + h) (y + fc)] = sin (a;?/) + liy cos (a;?/) + A;a; cos (xy) — ^^sin (xy) + 7tfc [cos (xy) — xy sin (a;^/)] - — - sin (xy) + • • •. 4. log(6-'+ e»-^) = log (^' + ^0 + ^g^' + ?(?+ ey + CHAPTER XII CHANGE OF THE VARIABLES IN DERIVATIVES 117. To express-^, -5, -^, ••• in terms of — , — ^, -5-5, ••■ da; dor dar dy dy' dy' This is changing the independent variable from, x to y. By (1), Art. 56, | = 1 (1) By (3), Art. 56, From (1), dy \dyj d^ .73.. .7 _72.. J j5.. .7.. Similarly, From (2), /d^Y da: d^a; (3) dy_ da; 1 "da; dy da;^" ^d^dy^ dxdx d dy dy dydx dx d^dy_ dydx' _d 1 _ ' dydx~ dy d^x df .dy) . d'y d'x df ' fdxy \dyj d^y_ da? _ d d'y _ dxda? d d^y dy dy da? dx Jdh d dFy _ \dy dyda? •y dx d'^x 7 dydf fdxY \dyj . d'y "da?' fd'xV dxd'x dydf \x\' 148 CHANGE OF THE VARIABLES IN DERIVATIVES 149 It is sometimes necessary in the derivatives, dy dPy d?y dx d^' d^' ■"' to introduce a new variable z in place of x or y, z being a given function of the variable removed. There are two cases, according as z replaces y or x. dy dV d^y . . , dz dh d^z -2, — 2. — 3. ... in terms of — , — -, — r, ■■- dx dx'' dx"' ' dx da^- da^ where y is & given function of z. By (3), Art. 56, dy^dyd^^ ■' ^ -" ' dx dz dx d'y _ d^fdy\dz ,dydh__ d^y/dzV dy d^ da^ dx\dz)dx dz dx' dz\dxj dz da?' Similarly, we find d^y _ d^yfdzY „ d^ dz_^ dy dh da?" d^\dx) dz^ dxdm? dz dv?' Similarly, — , —, •••, may be expressed in terms of z and x. It is to be noticed that in this case there is no change of the in- dependent variable, which remains x. For example, suppose y = z"'. Then ^ = 3z^—. dx dx dx^ \dxj doc!' ^^6f^Y+18z'^^,+ 3z'% doi? \dxj dxdx^ daf 150 DIFFERENTIAL CALCULUS 119. Second. To express^, ^, ^, ..., dx' da?' dar*' ' • ciii (aV d^t/ in terms of —,—,——,..., where x is a given function of z. dz dz^ dz^ This is changing the independent variable from x to z. By (3), Art. 56, # = ^' dy dx _dydz _ dz _ ~dzdx~ dx dz dfdy\ d'y_ da? d fdy\dz dz\dx) ~dz\dxjdx~ dx dz dx d^y dy d?x dz dz'' dz dz^ fdxY [dz] Similarly, higher derivatives may be expressed. In practice it is generally easier to work out each case by itself. For example, suppose x = ^. dy _ dy dz dx dz dx But dx o„2 dz z-^ dz~ '^~^- Hence dy_^^-2dy dx 3 dz (1) d^y _ d fdy\_ d fdy\dz da? dx \dxj dz \dxj dx ' From(l), i^f^V-r^-'-!-2'^^Y ^ ^' dz\dxj 3V dz' dz) Hence ^ = ^z-^^-2z-^^\ (2) dx^ ^\ dz" dz) ^> CHANGE OF THE VARIABLES IN DERIVATIVES 151 Similarly, ^^dMdz_ doc? dzyda^Jdx From (2), |f^A = If .-^ - 6 .- ^^ + 10 dzydx'J 9\ dz^ dz^ Hence ^ = i f,-e^_6«-'^ + 102 da? 27V d^ dz'^ ^-.dy dz EXAMPLES Change the independent variable from a; to t/ in the two following equations : 1. 3 f^Y- ^^- ^,%Y= Am ^ + ^-0 ydx^J dxd.v' dx\dx) ' • dy" dy' \ dx J\dafJ \ dx Jdx da? Ans. f^Y=f^ + a^^ \dfj \dy Jdf Change the variable from y to z in the two following equations : Ans. —- — 21—] = cos^z. dx' \dxj Change the independent variable from a; to z in the following equations : 5. ^^ + lf + y = 0, a^ = iz. Ans. z^^ + f + y=0. dx- xdx dz' dz 6. g^ + ^^^+ y , =0, a- = tan«. Ans. ^ + y = da? 1+xHx {l + x'f ' dz" ^ 152 DIFFERENTIAL CALCULUS 7. (2a;- iy^ + (2a! -1)^ = 22/, 2a; = l+e'. da? dx Ans. A^-12^ + 9^-y = 0. di^ dz^ dz 8. a;^^+6a!«5 + 9ar=5 + 3x^ + 2, = log*, x = e'. dx^ dar dar dx 120. Transformation of Partial Derivatives from Rectangular to Polar Coordinates. Given u = f(x, y), . to express — and — in terms of -^ and -^, where x, y, are rec- 5a; dy 5?- 66 tangular, and r, 6, polar coordinates. We have from (5), Art. 113, regarding m as a function of r and 6, du _ du dr , du dO^ ,-. , dx~d^-dx Bddx ^ ' Bu _ du dr , du dO ,„, dy dr dy 86 dy ' ' The values of -— , -— , ^— , — -, are now to be found from the rela- dx dy dx dy tions between x, y, and r, $. These are a; = rcosfl, y = rsmO (3) But in the partial derivatives -^, -^, and — , — , r and 6 are re- dx dy dx dy garded as functions of x and y. These are, from (3), 9^ = 3? + f, e = tan-'^. X CHANGE OF THE VARIABLES IN DERIVATIVES 153 Differentiating, we find — =- = cose, — = ^ = sin^, ox r ay r ^^_ y ^ sip^ ^9 ^ X _ cosg dx 3? + y^ r ' dy x' + y^~ r ' Substituting in (1) and (2), we have ^" „„„ a 5m sin —- = cos 6 , ax dr r 66 5" „„„ /] 5m sin ^ 3m ,., 9m • n Su , COS 5 9m /_, — =sine-— -I — . (5) ay dr r dO ^ 121. Transformation of 7— s + t-; fro™ Rectangular to Polar Co- aar dy^ ordinates. By substituting in (4), Art. 120, — for u, we have dx 9% _ 9 fdu\_ a d fdu\ sin 9 /9m'\ ..., daf~dx\dxj~ dr\dxj r de\dx) ^^ Differentiating (4), Art. 120, with respect to r, d /du\ cfi'^u sin 5 S^u , sin 9 9m ,„, — I — i = cosp (2) br\dx) dr" r drdd r' 39 ^ ^ Differentiating (4), Art. 120, with respect to 6, 9 /'9m\ /, d'n ■ n 9m sin Q d^u cos 6 du /„, — ( — l=cosS sm 6 . . . (3) de\dxj drdd dr r 86^ r 36 ^ ^ 154 DIFFERENTIAL CALCULUS Substituting (2) and (3) in (1), we have 5^i«_„„^2fl5^M 2 sin ^ cos 6 d'^u . sin^^5^w . sw^Odw ^^ COS (/ ~~" - —1— ~f~ di? d7^ r drde r' 86^ r dr 2 sin 6 cos du /^>, Similarly by using (5), Art. 120, instead of (4), we find d'u _ ■ 2 a d^u 2 sin 9 cos 6 d^u , cos^^ dhi , 005^9 du df~^^^ d? r drd9 i-^ d¥ r dr 2 sin 9 cos 9 du r" d9 Adding (4) and (5~> - - obtain ^'hi 3% _ dhi 1 9m 1 &^u da? dy" dr' r dr r^ d9^ .(5) CHAPTER XIII MAXIMA AND MINIMA OF FUNCTIONS OF TWO OR MORE INDEPENDENT VARIABLES 122. Definition. A function of two independent variables, f{x, y), is said to have a maximum value when x =a, y = b; when, for all sufB-ciently small numerical values of h and k, f(a,b)>f(a + h,b + k), (a) and a minimum value, when f{a,b)f{a + li,b), and for a minimum f{a,b) V) — ^j 'wlien x = a, y =b. ax Similarly, by letting 7j = in (a) and (6), we may derive -^ f{^, y) = 0, when x=a, y = b. dy These conditions for a maximum or minimum are necessary but not sufficient. As in the case of maxima and minima of functions of one variable, there are additional conditions involving derivatives of higher orders. These we shall give without proof, as their rigorous derivation is beyond the scope of this book. The conditions for a maximum or minimum value of u = f(x, y) are as follows : For either a maximum or minimum, 1^ = 0, and f^ = 0; (1) da; dy also ^SH^y d2u3Ju \dydx) dx'df ^' For a maximum, ^<0, and — <0 (3) ' dot? ' dy' ^ ' For a minimum, -— „ > 0, and — „ > 0. . . . . . (i) 03? dy" ' 124. Functions of Three Independent Variables. The conditions for a maximum or minimum value of u=f(x, y, %) are as follows: For either a maximum or minimum, dx ' dy ' dz ' and f^h^y^d^dju^ \dx dy) dx- dy" MAXIMA AND MINIMA OF FUNCTIONS For a maximum — < 0, and A < 0; flar 157 for a minimum, d'u >0, and A>0; d'u d\ d^u where A = dx'' dx dy dx dz d^u d'u B^u dx dy' dy"' dy dz d^u dhi d'u dx dz' dy dz' dz' EXAM PLES 1. Find the maximum value of M = 3 axy — 3? — ^. Here — = Say-S3?, ^=Sax-Zy\ dx dy ., d^u c 5^M „ d'u n dx' dy" dx dy Applying (1), Art. 123, we have ay — a? = 0, and aa — y^ = 0; whence x = Q, y = 0; 01 x = a, y = a. The values x = 0,y = 0, give ff'u _ ^ dhi _ f, d'u da? ' dy" dx dy which do not satisfy (2), Art. 123. Hence they do not give a maximum or minimum. 158 DIFFERENTIAL CALCULUS The values x = a, y = a, give dhi „ 9^M ^ dhi. o „ — = — 6 a, — t = — b a, = 6 a, 9ar Sy^ ox ay which satisf/ both (2) and (3), Art. 123. Hence they give a maximum value of u, which is a\ 2. rind the maximum value of xyz, subject to the condition ^;+g+^;=i (1) or W & rrom(l) g = l_^;_|!; and as xyz is numerically a maximum when cc^yV is a maximum, we put dxdy- \ a' W) From — = and — = 0, we find, as the only values satisfying 5a; dy (2), Art. 123, X — -^, y = , which give V3 V3 aa;^~ 9 ' dy^ 9 ' 5a;3y 9 MAXIMA AND MINIMA OF FUNCTIONS 159 As these values satisfy (2) and (3), Art. 123, it follows that xyz is a maximum when „ a 6 c V3 V3 V3 The maximum value of xyz is-" 3V3 3. Find the values of x, y, z that render x^ + y'' + z^ + X — 2 z — ayy a minimum. . 2 1^ Ans. x=—-, «= , « = 1. 3 3 4. Find the maximum value of (a - x)(a - y)(x + y - a). Ans. —. 5. Find the minimum value of /^ — ax — hii Ans 3? + xy + y'^ — ax—hy. Ans. —(ah — a? — V). 6. Find the values of x and y that render sin X + sin y + cos (a; + y) a maximum or minimum. Ans. A minimum, when x = y = — ; a maximum, when a; =w = 5. ' ^ 6 7. Find the maximum value of aj^-l- 2/^ + 1 8. Find the maximum value of ':^'^'^, subject to the condition 2x-\-Zy-\-'^z = a Ans. f- X y z a c xyz = dbc. Ans. 3. Find the minimum value oi - + •- + ~, subject to the condition a h c 160 DIFFERENTIAL CALCULUS 10. Divide a into three parts such that their continued product may be the greatest possible. Let the parts be x, y, and a — x — y. Then u = oey (a — x — y), to be a maximum. — =ay-2xy-y^ = 0, — = ax-x'-2xy = 0. dx dy These equations give x = y = -. Hence a is divided into equal parts. Note. — When, from the nature of the problem, it is evident that there is a maximum or minimum, it is often unnecessary to consider the second derivatives. 11. Divide a into three parts, x, y, z, such that x'^y^'z' may be a maximum. A g _ y _g _ a m n p m-\-n-\-p 12. Divide 30 into four parts such that the continued product of the first, the square of the second, the cube of the third, and the fourth power of the fourth, may be a maximum. Ans. 3, 6, 9, 12. 13. Given the volume a' of a rectangular parallelopiped ; find when the surface is a minimum. Ans. When the parallelopiped is a cube. 14. An open vessel is to be constructed in the form of a rec- tangular parallelopiped, capable of containing 108 cubic inches of water. What must be its dimensions to require the least material in construction ? Ans. Length and width, 6 in. ; height, 3 in. 15. Find the coordinates of a point, the sum of the squares of whose distances from three given points. MAXIMA AND MINIMA OF FUNCTIONS 161 is a minimum. . 1 / , , . 1 / , , v Ans. -{Xi + X2 + Xs), g (2/1 + 2/2 + 2/3)) the centre of gravity of the triangle joining the given points. 16. If X, y, z are the perpendiculars from any point P on the sides a, 6, c of a triangle of area A, find the minimum value of x^ + 2/^ + z^. 4A^ Ans. a? + W + (^ 17. Knd the volume of the greatest rectangular parallelopiped that can be inscribed in the ellipsoid, t + t + l = l. Ans. ^. Of W (? 3V3 18. The electric time constant of a cylindrical coil of wire is mxyz u = , ax + hy + cz where x is the meaii radius, y is the difference between the internal and external radii, 2 is the axial length, and m, a, b, c are known con- stants. The volume of the coil is nxyz = g. Find the values- of x, y, z which make u a minimum if the volume of the coil is fixed ; also the minimum value of u. Ans. ax=by = cz=^S. u = ^\l-S^ V n 3 yiabm' CHAPTER XIV CURVES FOR REFERENCE We give in this chapter representations and descriptions of some of the curves used as examples in the following chapters. RECTANGULAR COORDINATES Y 125. The Ci^oid, r 3? 2a- This curve may be constructed from the circle ORA (radius a) by drawing any oblique line OM, and making PM= OB. The equation above may be easily obtained from this construction. The line AM parallel to F is an asymp- tote. The polar equation of the cissoid is r=2a sin 6 tan 9. 162 126. The Witch of Agnesi, y — CURVES FOR REFERENCE 8 a' x^ + 4 a^ 163 This curve may be constructed from the circle OBA (radius, a) by- drawing any abscissa MR, and extending it to P determined by ORN, by the construction shown in the figure. The equation above may be derived from this construction. The axis of X is an asymptote. 127. The Folium of Descartes, 3? + if — ^ axy = 0. The point A, the vertex of the loop, is f3a 3a\ The equation of the asymp- tote MN is x + y+a = 0. The pplar equation of the folium is 3 a tan d sec 6 ''" l + tatfe a a 164 DIFFERENTIAL CALCULUS 128. The Catenary, y = -{er + e "). This is the curve of a cord or chain suspended freely between two points. 129. The Parabola, referred to Tangents at the Extremities of the ill Latus Rectum, a;^ + y^ = « ■ OL = OL The line LL' is the latus rectum ; its middle point F, the focus ; OFM, the axis of the parabola ^ A the middle point of OF, the vertex. CURVES FOR REFERENCE 165 130. The curve a"~^y = x", wHere one coordinate is proportional to the nth power of the other, is sometimes called the parabola of the nth degree. If w = 3, we have the Cubical Parabola, a'y = a?. If 71= -, we have the Semicubical Parabola, LI a^y= a; 2, ay^ = a?. 166 DIFFERENTIAL CALCULUS 131. The Two-arched Epicycloid. X = -^cos <^ — -COS 3 i}>j y= — sin<^--sin3<^. 132. The Hypocycloid of Four Cusps sometimes called the Astroid, 2 2 2 x^ + y^ = a^- This is tlie curve de- scribed by a point P in the circumference of the circle PE, as it rolls within the circumfer- ence of the fixed circle ABA', whose radius a is four times that of the former. The equation above may be given in the form a;=acos^<^, y=a sin^ff- CURVES FOR REFERENCE 167 133. The Curve, The equation is the same as that of the ellipse with the exponent of the second term changed from 2 to|. 134. The Curve, ay = aV-x\ Y POLAR COORDINATES 135. The Circle, r = a sin 6. The circle is OPA (diameter, a) tangent to the initial line OX at the origin 0. 168 DIFFERENTIAL CALCULUS 136. The Spiral of Archimedes, r = ad. In this curve r is proportional to 6. Lay- ing off r=OA, when e = 2iT, then OPi = {OA, OP, = \OA, OPs = ^OA, OP, = iOA, 0B = 2 0A, 00 = 3 0A. The dotted portion corresponds to negative values of 0. 137. The Hyperbolic or Reciprocal Spiral, rO = a. In this "Curve r varies inversely as 0. The line MN is an asymptote, which the curve approaches, as 6 approaches zero. Since r=0 only when 6=00 , it fol- lows that an in- finite number of revolutions are necessary to reach the origin. CURVES FOR REFERENCE 169 138. The Logarithmic Spiral, r = e"* Starting from A, where ^=0 andr=l, r increases "with 6 ; but if we suppose 6 negative, r decreases as 6 numerically in^ creases. Since r=0 only when 6=—oo, it follows that an infinite number of retrograde revolu- tions from A is re- quired to reach the origin 0. A property of this spiral is that the radii vectores OP, OPj, OP2, ■ make a constant angle with the curve. 139. The Parabola, Origin at Focus, r(l — eos6) = 2a. The initial line OX is the axis of the parabola; the origin is the focus; LL', the latus rectum. 140. The Parabola, Origin at Vertex (see preceding figure), r sin 6 tan 6 = 4a. The initial line is the axis AX; the origin is the vertex A. 170 DIFFERENTIAL CALCULUS 141. The Cardioid, r = (((1 — cos 6). This is the curve described by a point P in the circum- ference of a circle PA (di- ameter, a) as it rolls upon an equal fixed circle OA. Or it may be constructed by drawing through 0, any line OB in the circle OA, and producing OE to Q and Q', making RQ=BQ'=OA. The given equation fol- lows directly from this con- struction. 142. The Equilateral Hyperbola, r^ cos 2 ^ = a'. The origin is the centre of the hyperbola, and the in- itial line OX is the trans- verse axis. If or is taken as the initial line, the equation of the hyper- bola is ?-2 sin 2 e=a\ CURVES FOR REFERENCE 171 143. The Lemniscate referred to OA (see preceding figure), r^ = a' cos 2 $. This is a curve of two loops like the figure eight. It may be defined in connection with the equilateral hyperbola, as the locus of P, the foot of a perpendicular from on PQ, any tangent to the hyperbola. The loops are limited by the asymptotes of the hyperbola, making TOX=T'OX= 45°. OA = a. The lemniscate has the following property : If two points, F and F', called the foci, be taken on the axis, such that OF=OF'=- V2 then the product of the distances P'F, P'F', of any point of the curve from these fixed points, is constant, and equal to the square of OF. If or ' is taken as the initial line, the equation of the lemniscate is r'^a^ sin 2 $. 144. The Four-leaved Rose, r = a sin 2 ^. 172 145. The Curve, r=a sin' DIFFERENTIAL CALCULUS 6 CHAPTER XV DIRECTION OF CURVES. TANGENTS AND NORMALS We liave seen in Art. 17 that the derivative at any point of a plane curve is the slope of the curve at that point. We will nov7 con- sider some further applications of differentiation to curves. 146. Subtangent, Subnormal, Intercepts of Tangent. — Let PT be the tangent, and FN the normal, to a curve at the point P, whose ordinate is y = PM. Then MT is called the subtangent, and MN the subnormal, corre- sponding to the point P. To find expressions for these quantities : Let 4) donote the angle PTX, the in- clination of the tan- gent to OX. By Art. 17, tanP-.rX=tan = -^ = dx dx Intercept of tangent on OX=OT= OM- TM=x-y—- Intercept of tangent onOT=Or=PS- PM= x tan ^-y. But as OT' is negative, we have Intercept of tangent on Y= y - 173 -a; tan', and let / be the angle TPT' between the tangents. Then /= <^' - <^ and tanJ= tan-^'-tan.^^ .^. l+tan<^'tan<^ ' From the equations of the given curves find the coordinates of the point of intersection P; then using these equations separately, find by tan = — the values of tan (p dx and tan <^' for the, point P. Substituting in (1) gives tan I. For example find the angle at which the circle (2) T 3^ + 2/2 = 13, . . . . intersects the parabola 2y' = 9x (3) The intersection P of (2) and (3) is found to be (2, 3). / Differentiating (2), ^ = -^ = -2forP,tan<^ = -|. dx y 3 3 From (3), dy^9_^3 dx 4?/ 4 for P, tan <^' = - Substituting in (1), tan / = ^, /= 70° 33', DIRECTION OF CURVES. TANGENTS AND NORMALS 175 EXAMPLES 1. Find the direction at the origin of the curve, (a* -h*)y = x(x- a)* — b*x. Ans. 45° with OX. What must be the relation between a and h, so that it may be parallel to OX at the point x = 2a'! Ans. 3 a^ = V. 2. Find the points of contact of the two tangents to the curve, 6?/ = 2a^ + 9a^-12a; + 2, parallel to the tangent at the origin to the curve, y^ + ay = 2ax. Aiis. [ 1, - j, (— 4, II)- 3. Find the subtangents and subnormals in the parabolas, 7/^ = 4 ax, and a^ = 4 ay. Ans. Subtangents, 2 x, -; subnormals, 2 a,r— ^• 4. Find the subtangent and subnormal in the cissoid (Art. 125), y' = , at the point (a, a). Ans. -, 2 a. 2a — X 2 5. Show that the sum of the intercepts of the tangent to the parabola (Art., 129), x'+ y^= (V, is equal to a. 6. Show that the area of the triangle intercepted from the co- ordinate axes by the tangent to the hyperbola, 2 ««/ = a?, is equal to a^. 7. Show that the part of the tangent to the hypocycloid (Art. 132), a;^ +2/'= a', intercepted between the coordinate axes, is equal to a. 8. At what angle do the parabolas, y^ = ax and a^ = Say intersect? 3 Ans. At (0, 0), 90°: at another point, tan-^ -• 5 176 DIFFEREITTIAL CALCULUS 9. At what angle does the circle, a^ + 2/^ = 5 a;, intersect the curve, 3y = 7a^ — l, at their common point (1, 2) ? Ans. 45°. 10. Show that the ellipse and hyperbola, 7 + 2" ' 3 2~ ' intersect at right angles. 11. Find the angle of intersection of the circles, «? + y''-x + 3y + 2 = 0, x' + y^-2y = 9. Ans. tan"'-. 12. Show that the parabola and ellipse, y^ = ax, 2 x^ + y^:= b% intersect at right angles. 13. Show that the parabolas, y^ = 2ax + a?, and a? = 2'by + b^, intersect at an angle of 46°. 14. Find the angle of intersection of the parabola, 8 a' 0^ = 4: ay, and the witch (Art. 126), y = ar*+4a^ Ans. tan-i3=71°34'. 15. Find the angle of intersection between the parabola, 2/2 = 4 ax, and its evolute, 27 ay^ = 4 (a; - 2 af. (See Fig., Art. 167.) Ans. tan~* V2 148. Equations of the Tangent and Normal. Having given the equation of a curve y=f(x), let it be required to find the equation of a straight line tangent to it at a given point. DIRECTION OF CURVES. TANGENTS AND NORMALS 177 Let (a;', y') be the given point of contact. Then the equation of a straight line through this point is y — y' = m(x-x'), (1) in which x and y are the variable coordinates of any point in the straight line ; and m, the tangent of its inclination to the axis of X. But since the line is to be tangeut to the given curve, we must have, by Art. 17, m = tan (f> = dy dx' -^ being derived from the equation of the given curve y =f(x), and applied to the point of contact (a;', y'). If we denote this by -^ , we have, substituting m = -^ in equation (1), '^^' '^^' y-y'^^,ix-x'), (2) for the equation of the required tangent. Since the normal is a line through' (w', t/') perpendicular to the tangent, we have for its equation dx' For example, find the equations of the tangent and normal to the circle af + y^ = a^, at the point (x', y'). Here, by differentiating af + y^^ a% we find -^ = , from which -^ = . dx y dx' y' Substituting in (2), we have y — y'==--(x — x'), y as the equation of the required tangent. 178 DIFFERENTIAL CALCULUS It may be simplified as follows : yy' — y'"^ = — ^^' + x' ^ xx' + 2/2/'= k'^ 4- 2/' ^ = c^. The equation of the normal to the circle is found from (3) to be „' X which reduces to x'y = y'x. EXAMPLES Find the equations of the tangent and normal to each of the three following curves at the point (x', y') : 1 . The parabola, y'' = A: ax. Ans. yy' =2 a(x + x'), 2a(y — y') +y'(x — x') =0. 2. The ellipse, ^„ + f' = l. Ans. ^ + 2^ = 1, Vx'(y — y') = a^y'(x — x'). 3. The equilateral hyperbola, 2xy = a^. Ans. xy' + yx' =:a^, y'(y—y') = x'(x — x'). 4. Find the equation of the tangent at the point (x',y') to the ellipse, 3x'-4:xy + 2r/+2x = 2. A71S. 3 a;a;' + 2 yy'— 2 (x'y + y'x) + a; + «' = 2. 5. Find the equations of tangent and normal at the point (x', y') to the curve, a^ = a'y^. Ans. ^-^ = 3, 2xx' + 5yy' = 2x'^ + 5y'\ x' y' DIRECTION OP CURVES. TANGENTS AND NORMALS 179 6. In the oissoid (Art. 125), f = --^, find the equations of the tangent and normal at the points whose abscissa is a. Ans. At (a,a), y = 2x—a, 2y-\-x = 2,a. At (a, — a), y + 2a; = a, 2y = x — Za. 1. In the witch (Art. 126), y = ^J^' find the equations of the tangent and normal at the point whose abscissa is 2 a. Ans. x + 2y = 4:a, y = 2x—3a. 8. Find the equation of the tangent at the point (x', y') to the curve, x^y + xy'^ = a '. Ans. xy\2 x'+y') + yx'{2 y' + a;') = 3 a\ Find the equations of tangent and normal to the three following curves : 9. a;^+2/'=3a.ry(Art. 127),at('^,|^\ Ans. x+2/=3a, x=.y. 10. x + y = 2e'-y,a.t(l,l). Ans. 3y = x + 2, 3x + y = 4:. 11. /'^Y + r|y= 2, at (a, b). Ans. - + 1 = 2, ax-by = a'~b\ 12. Find the equations of the two tangents to the circle, x^ + y^ — 3y = 14:, parallel to the line, 7 y = 4 a; + 1. Ans. 7y = 4:X + i3, 7y = ix — 22. 13. Find the equations of the two normals to the hyperbola, 4 a^ - 9 2/2 + 36 = 0, parallel to the line, 2y + 5x = 0. Ans. 82/ + 20a;=±65. 149. Asymptotes.* When the tangent to a curve approaches a limiting position, as the distance of the point of contact from the origin is indefinitely increased, this limiting position is called an * The limits of this work allow only a brief notice of this subject. 180 DIFFERENTIAL CALCULUS asymptote. In other words, an asymptote is a tangent which passes within a finite distance of the origin, although its point of contact is at an infinite distance. We have found in Art. 146, for the intercepts of the tangent on the coordinate axes, Intercept on 0X= x — y — , Intercept on 0Y=: y — x-^. ay cios If either of these intercepts is finite for a; = oo, or ?/ = oo, the cor- responding tangent will be an asymptote. The equation of this asymptote may be obtained from its two intercepts, or from one intercept and the limiting value of -^. dx Let us investigate the conic sections with reference to asymptotes. (1) The parabola, w^ = 4 aa;, ^ = ?_^ . dx y Intercept on OX = x —y — = x — ^ = — x, dy 2 a Intercept on OY = y — x-J = y = ^. dx 2/2 When a! = 00, y-=ai, and both intercepts are also infinite. Hence the parabola has no asymptote. (2) The hyperbola, ^-g = l, f^ = ^. a^ ¥ dx a'y Intercept on OX = — , Intercept on OY = . X y These intercepts are both zero when a; = oo, and there is an asymptote passing through the origin. To find its equation, it is necessary to find the limiting value of -^, when a; = oo. Hence di/_ _Vx d'y = ± bx — ± 6 1 dx "Va^- -a' "V- a' 'a? dx a when a; =00. DIRECTION OF CURVES. TANGENTS AND NORMALS 181 There are then two asymptotes, whose equations are y = ±-x. a (3) The ellipse, having no infinite branches, can have no asymptote. 150. Asymptotes Parallel to the Coordinate Axes. When, in the equation of the curve, a; = oo gives a finite value of y, as y = a, then 2/ = a is the equation of an asymptote parallel to OX. And when y = oo gives x = a, then a; = a is an asymptote parallel to or. 151. Asymptotes by Expansion. Trequently an asymptote may be determined by solving the equation of the curve for x or y, and expanding the second member. For example, to find the asymptotes of the hyperbola a- b' , = ±_(,-_a.).= ±_^l__j.= ±_(^l___. As X increases indefinitely, the curve approaches the lines « = ± — , the asymptotes. a EXAMPLES Investigate the following curves with reference to asymptotes r 1. y — ?- — . Asymptote, 2/ = a;. 2. f=^&x^-3?. Asymptote, a; + y = 2. a? 3. The cissoid (Art. 125) y'^ = Asymptote, a; = 2 a. ^ 2 a — a; 182 DIFFERENTIAL CALCULUS 4. af' + 2/' = a^ Asymptote, x +y = 0. 5. (x — 2 a)y^ = a? — a^. Asymptotes, x = 2a, x + a = ± y. 6. a? ■\-y^ = 3 axy (Art. 127). Asymptote, x + y+a = Q. (Substitute y = vxm the given equation and in the expressions for the intercepts.) 152. Direction of Curve. Polar Coordinates, In this case the angle OPT between the tangent and the radius vector may be most readily obtained. Denote this angle by ^. Let r, 0, be the coordi- nates of P; r+\r,e+^e, the coordinates of Q. Draw PB perpendicular to OQ. PR Then tanPQ^ = i^ = . ^_ BQ r+Ar-rcosA6l ^,. , o ,. ^^2 AS 2 .sin A& ■ A6» sm— - Ar , . A0 2 \-r sm — Afl 2 A^ 2 Now let A5 approach zero ; the point Q approaches P, and the angle PQB approaches its limit i//. T Hence tan i// = Lim^j^otan PQP = — (1) The inclination <^ of the tangent to OX may be found by <#. = ./'+ 61 (2) DIRECTION OF CURVES. TANGENTS AND NORMALS 183 153. Polar Subtangent and Subnormal. If through 0, NT be drawn per- pendicular to OP, or is called the polar subtangent, and ON the polar subnorTfial, corresponding to the point P. 0T= OP tan OPT; that is, Polar subtangent = r tan tl/ = — . 0N= OP cot PNO; that is, Polar subnormal = r cot ij/ = dr 154. Angle of Intersection. Suppose the two curves intersect at P, and have the tangents PT and P2". OPT=^, OPT' = f. Then the angle of intersection, and tan/^^^'^'^'-t^"'/'. (1) 1 + tan i/r' tan i/r ^ By this formula the angle of inter- section may be found in polar coordi- nates, in the same way as by (1), Art. 147, in rectangular coordinates. For example, find the angle of intersection between the curves r = asm2d, (2) and r — a cos 20. (3) From (2) and (.3) we have for the intersection tan20 = l. 184 DIFFERENTIAL CALCULUS From (2), tan = 2 6. 2. In the logarithmic spiral (Art. 138), )■ = e"^, show that \j/ is constant. 3. In the spiral of Archimedes (Art. 136), r = ad, show that tan \ji=d; thence find the values of i//, when 6 = 2t!- and 4 tt. Ans. 80° 57' and 85° 27'. Also show that the polar subnormal is constant. 4. The equation of the lemniscate (Art. 143) referred to a tangent at its center is r^ = a' sin 2 6. Find \p, 4>, and the polar subtangent. Ans. il/ = 26; ct> = 3 6; subtangent = a tan 2 6 Vsin 2 6. 5. In the cardioid (Art. 141), r = a(l —cos 6), find <^, kj/, and the polar subtangent. o in ^ ft ft Ans. = -x- ; "A = 7 ; subtangent = 2 o tan - sin^ -. A ^ Li Z 6. Find the area of the circumscribed square of the preceding cardioid, formed by tangents inclined 45° to the axis. 27 - DIRECTION OF CURVES. TANGENTS AND NORMALS 185 3 a tan 6 sec 7. In the folium of Descartes (Art. 127), r= show that tan <^ 1 + tan^e tan* - 2 tan e 2tan''e-l 8. Find the area of the square circumscribed about the loop of the folium of the preceding example. Ans. 2A/2al 9. Show that the spiral of Archimedes (Art. 136), r = ad, and the reciprocal spiral (Art. 137), 7-6 = a, intersect at right angles. 10. Show that the cardioids (Art. 141), r = a (1 — cos 0), r = b(l + sin 6), intersect at an angle of 45°. 11. Show that the parabolas (Art. 139), r = m sec' -, r = n cosec- - 2 ^ intersect at right angles. 12. Find the angle of the intersection between the circle (Art. 135), 9- = a sin 0, and the curve (Art. 144), r = asin26. Ans. At origin 0° ; at two other points, tan"' 3V3 = 79° 6'. 13. Find the angle of intersection between the circle (Art. 135), r = 2acose, and the cissoid (Art. 125), r = 2 a sin 6 tan 0. Ans. tan"' 2. 14. At what angle does the straight line, rcos6 = 2a, intersect the circle (Art. 136), r = 6 a sin 61 ? ^,jg_ ^^n-' -. 4 15. Show that the equilateral hyperbolas (Art. 142), r^ sin 2 6 = a% 1^ cos 2 6 = 6^, intersect at right angles. 186 DIFFERENTIAL CALCULUS 16. Find the angle of intersection between the circles r = asiQO + bcosd, r = acos9 + b sin 6. Ans. tan 2 ah ' 17. Find the angle of intersection between the lemniseate (Art.143), 7^=: a? sin 2 6, and the equilateral hyperbola (Art. 142), r^sin 2 6 = ¥. Ans. 2 sin~^ — a 155. Derivative of an Arc. Rectangular Coordinates. Let s denote the length of the arc of the curve measured from any fixed point of it. Then We have s = arc AP, As = arc PQ. secQPi? = ||. Now suppose Ax to approach zero, and consequently the point Q to approach P. Then Lim sec QPi2=sec TPE=sec- Here ds=V(dxy+(dyy, ds dx -v^HiJ 156. Derivative of an Arc. Polar Coordinates. Trom the figure of Art. 152, we have, as A9 approaches zero. sec i/r = Lim sec PQR = Lim — ^ = Lira — ^ = Lim — — RQ RQ As RQ As As A^ Ar + 2rsin^^ ^ sin"^^ 2 Ar . ^e A9 2 A^ ■^sin-^" RQ Hence ds T ■ As dO ds sec d/ = Lim --— = — = --. ^ RQ dr dr dd ds ,3 =— ^= Vl+tan^^ ds de drdd =Vi+<'' \dr_ '~'cb-dti~'\ \da (1) (2) (3) 188 DIFFERENTIAL CALCULUS It may be noticed that these relations (1), (2), and (3), are cor- rectlj' represented by a right triangle, whose hypothenuse is ds, sides dr and rd$, and angle between dr and ds, ij/. Here ds = ■y/{dry^-{rddf, „ath.„.. i-vN+.'0" o, |.V"+(i; CHAPTER XVI DIRECTION OF CURVATURE. POINTS OF INFLEXION 157. Concave Upwards or Downwards. A curve is said to be con- cave upwards at a point P, when in the immediate neighborhood of P it lies wholly above the tangent at P, as in the first figure below. Similarly, it is said to be concave downwards, when in the immediate neighborhood of P it lies wholly below the tangent at P, as in the second figure below. It will now be shown that when the equation of the curve is in rectangular coordinates, the curve is concave upwards or downwards, according as —- is positive or negative, asc Suppose —^>0, that is, — ( — ] > 0: in other words, the derivative dor dx\dxj of the slope is positive. Then by Art. 21 the slope increases as x increases. This case is illustrated in the first figure above, where the slope 'evidently increases as we pass from P^ to Pj- The curve is then con- cave upwards. d'v But if -~ < 0, it follows that the slope decreases as x increases. dx' 189 190 DIFFERENTIAL CALCULUS We then have the case of the second figure, where the slope de- creases as we pass from Pj to P^. The curve is then concave down- wards. 158. A Point of Inflexion is a point P where -^ changes sign, dx' the curve being concave upwards on one side of this point, and con- cave downwards on the other. This can occur, provided -^ and dx —^ are continuous, only when da^ ^ da? :0. (1) d'y But if ^ and ^, are infinite, we dx daf may have a point of inflexion when — 2 = ( da? It is evident that the tangent at a point of inflexion crosses the curve at that point. For example, find the point of inflexion of the curve 2y = 2-8x + 6x'-a?. Here ^ = 3(2-x). dx' ^ ' Putting this equal to zero, we have for the required point of in- flexion, a; = 2. If a;<2, f^,>0;andif «>2,^<0. Hence the curve is concave upwards on the left, and concave down- wards on the right, of the point of inflexion. DIRECTION OF CURVATURE. POINTS OF INFLEXION 191 EXAMPLES Find the points of inflexion and the direction of curvature of the five following curves: 1. 2/=(a^-l)2. Ans. x=± — -; concave downwards between these points, con- V3 cave upwards elsewhere. 2. y = x*-16a^ + 4:2x'-28x. Ans. a; = 1 and x=7; concave downwards between these points, concave upwards elsewhere. 3. a*y =x(x — ay + a*x. Ans. a;=— ; concave downwards on the left of this point, con- o cave upwards on the right. 4. The witch (Art. 126), y=-^ af + ia' Ans. ( ± — -, — - j ; concave downwards between these points, concave upwards outside of them. 5. The curve, y = A71S. ( —3a, ' ], (0, 0), 1 3 a, — — ] ; concave upwards on the left of first point, downwards between first and second, upwards between second and third, and downwards on the right of third point. 192 DIFFERENTIAL CALCULUS Find the points of inflexion of the following curves : ix V4-4 6- 2' = :T-r7' ^ns. a; = 0and±2V3. 7. 2/ = —^^-^ — Ans. x = -2a. (x — ay 8. y = {a:F + x) e". Ans. a; = and x = 3. 9. 3/ = e-" — e-'^ Ans. a, = 2(loga-logb)_ a — b 10. [^X + (yy=l. Ans. x=±-^ 2 w. 11. ay = aV - x" (Art. 134). ^ws. «; = ± ^ "^27 - 3 ■v'33. CHAPTER XVII CURVATURE. RADIUS OF CURVATURE. EVOLUTE AND INVOLUTE 159. Curvature. If a point moves in a straight line, the direc- tion of its motion is the same at every point of its course, but if its path is a curved line, there is a continual change of direction as it moves along the curve. This change of direction is called curvature. We have seen in the preceding chapter that the sign of the second derivative shows which way the curve bends. We shall now find that the first and second derivatives give an exact measure of the curvature. The direction at any point being the same as that of the tangent at that point, the curvature may be measured by comparing the linear motion of the point with the simultaneous angular motion of the tangent. 160. Uniform Curvature. The curvature is uniform when, as the point moves over equal arcs, the tangent turns through equal angles. The only curve of uniform curvature is the circle. Here the meas- ure of curvature is the ratio between the angle described by the tan- gent and the arc described by the point of contact. In other words, it is the angle described by the tangent while the point describes a unit of arc. Suppose the point P to move in the circle AQ. Let s denote its distance AP from some initial position A, and <^ the angle PTX made by the tangent PT with OX. Then as the point moves from P to Q, s is increased by PQ = As, and <^ by the angle QRK= A<^. As the point describes the arc As, the tangent turns through the angle A dd} As ds CURVATURE. RADIUS OF CURVATURE 195 162. Circle of Curvature. A circle tangent to a curve at any point, having its concavity turned in the same direction, and having the same curvature as that of the curve at that point, is called the circle of curvature ; its radius, the radius of curoature ; and its centre, the centre of curvature. The figure shows the circle of curvature MPN for the point P of the ellipse. C is the centre of curvature, and GP the radius of curvature. It is to be noticed that the circle of curv- ature crosses the curve at P. This can be easily proved. At P the circle and ellipse have the same curvature, but as we go towards Pi, the curvature of the ellipse increases, while that of the circle continues the same. Hence on the right of P the circle is outside of the ellipse. Moving from P to Pj , the curvature of the ellipse decreases, and therefore on the left of P the circle is inside of the ellipse. So in general the circle of curvature crosses the curve at the point of contact. 196 DIFFERENTIAL CALCULUS The only exceptions to this rule are at points of maximum and minimum curvature, as the vertices A and B of the ellipse. As we move from A along the curve in either direction, the curva- ture of the ellipse decreases ; hence the circle of curvature at A lies entirely within the ellipse. Similarly it appears that the circle of curvature at B lies entirely without the ellipse. 163. Radius of Curvature. The curvature of the circle of curvar ture being that of the given curve, is equal to ^ (Art. 161). If we ds denote the radius of curvature by p, then by Art. 160, ds ,., '^di (^) ds_ To obtain p in terms of x and y, we may write (1), p = — = _ ? • d~d' dd From (3), Art. 156, From (2), Art. 152, = ^ + ^' ■■■fe = '^fe 198 DIFFERENTIAL CALCULUS From (1), Art. 152, dO = tan-[i: dr • fdrS ^dV d^ \de) ^dO^ " --^(IJ Substituting, P'HS)"] 1 r" -KIT- dV dSP (1) EXAMPLES Find the radius of curvature of the following curves : 1.2/ = (x-iy(x-2), at (1, 0) and (2, 0). Ans. p = ^ and -i. 2. 2/ = log CK, when a; = |. ^ws. p = 2||. 3 3. The cubical parabola (Art. 130), a^y = af'. ^)is. p = ^^ — "1 ^ ^ • 2 4. The parabola, y^ = iax. Ans. p = ^ "^ — i- • Find the point of the parabola where p = 54 a. Ans. a; = 8 a. 3 5 The equilateral hyperbola, 2xy = a-. Ans. p = ^^ — / ' ■ CURVATURE. RADIUS OF CURVATURE 199 6. The ellipse, ^ + ^^ = 1. Ans. p = («y + y«^)l What are the values of p at the extremities of the axes ? Ans. — and — . a b 7. Show that the radius of curvature of the curve, ^ + y^+ 10 aj — 4 2/ + 20 = is constant, and equal to 3. Find the radius of curvature of the following curves : 8. 2/+ log (1 - a^) = 0. Ans. p = 2^^'- 9. smy = e''. Ans. p = e~''. 10. The catenary (Art. 128), 2/ = ^ (e» + e^«). Ans. p = ^. Ji CI 11. The hypocycloid (Art. 132), x^ +y^ = a*. A71S. p = 3 (axy)^- 12. The curveay = aV-a!« (Art. 133), at the points (0, 0) and a, 0). Ans. p = %nd p = a. 13. The cycloid, a; = a((9- sin 0), y = a(l—cose). Ans. p = 4asin-' 14. Show that the radius of curvature of the logarithmic spiral (Art. 138), r = e"^, is proportional to r. P = '■ Vl + a-. 15. Show that the radius of curvature of the curve, r = a sin ^ 4- & cos 0, is constant. P = 2 v a^ + b^- 16. The spiral of Archimedes (Art. 136), r^aO. Ans. p = 3 T o 2 • 200 DIFFERENTIAL CALCULUS 17. The cardioid (Art. 141), r = a (1 - cos ^). Ans. 18. The curve, r=:a sin^ - (Art. 145). o 19. The parabola (Art. 139), r = a sec' ^- Ans. p = 2a sec' -ar. 9 A 3 • oO Ans. p=-asin^-- 20. The lemniscate (Art. 143), 7^ = a' cos 2 6. Ans. p = - 3j- 165. Coordinates of the Centre of Curvature. Let x, y be the co- ordinates of P, any point of the curve AB, and G the corresponding centre of curvature. CP is then the radius of curvature, and is normal to the curve. Draw also the tangent PT. Then CP=p; angle PCE = PTX = 4,. Let a, p, be the coordinates of C. OL = OM- BP, LC=MP+BC; that is, a =x — p sin <^, P = y + pcos(j>. (1) To express a and j3 in terms of x and y, we have, by (2), Art. 155, and (1), (2), Art. 163, dy ■ ,_ds dy _dy _ dy dx dx dtp ds dcj> dx dfji 1 + dyV dx) d?i , ds dx dx p cos

ds dfj} Hence a = x — dy dx \dxj dx^ ■ cVy dx^ > ;8 = 2/ + dx^ (2) CURVATURE. RADIUS OF CURVATURE 201 166. Evolute and Involute. Every point of a curve AB has a Thus, -'D -^2) "■> etc., have for corresponding centre of curvature their respective centres of curvature d, C^, Cg, etc. The curve HK, -which is the locus of the centres of curvature, is called the evolute of AB. To express the inverse relation, AB is called the involute of HK. 167. To find the Equation of the Evolute of a Given Curve. By (2), Art. 165, a and /3, the coordinates of any point of the required evolute, may be expressed in terms of x and y, the coordinates of any point of the given curve. These two equations, together with that of the given curve, furnish three equations between a, (3, x, and y, from which, if x and y are eliminated, we obtain a relation between a and ji, which is the equation of the required evolute. For example, find the equation of the evolute of the parabola Here 2/^ = 4 ax. dx ' d3? 2 Substituting in (2), Art. 165, we have 2x^ Eliminating x, we have for the equation of the evolute, This curve is the semicubical parabola (Art. 130). The figure shows its form and position. F is the focus of the given parabola. • OC=2a = 2 0F. 202 DIFFERENTIAL CALCULUS As another example, let us find the equation of the evolute of the ellipse, o 2 -+^ =1. dy __ &^ d^y __ i* dx a'y dx' a'y (Art. 66) Substituting in (2), Art. 16^, a' ' l^- b' To eliminate x and y be- tween these equations and that of the ellipse, we find • b'' ¥ f bl_ ■b'' x' y'^_ iaay + (bpy _-, giving, for the equation of the evolute, (aay^+(&;8)*=K-&-)'. The evolute is EF'E'FE. E is centre of curvature for ^; C for P; FtoT B; E' for A'; F' for B'. In the figure F and F' are outside the ellipse, but if the eccentric- ity is decreased, so that a, P = y + pcoa^. CURVATURE. RADIUS OF CURVATURE 203 Differentiating with respect to s, da ds d§_ ds dx ds dp ds ■ p cos -2 ds (1) ds dp ds cos^— psinc^-^- ds (2) Substituting in (1), ds J , dx p = — and cos.^= — , d(j> ds (Art. 155), two terms cancel each other, giving da ds ■^sin,/,. Similarly in (2), ^ = *- and sin <^ = ^ (Art. 156), giving d ds ^ = ^cos<^ ds ds Dividing (4) by (3), ^ = aa d/3 tan<^ (3) (4) (5) But -!- is the slope of the tangent to the evolute at any point Oi, (see fig., Art. 166), and tan cji the slope of the tangent to the involute at the corresponding point P^. Since by (5) one is minus the recip- rocal of the other, these tangents are perpendicular to each other. In other words, a tangent to the evolute at any point Oi is Ci Pi, the normal to the involute at Pi. 204 DIFFERENTIAL CALCULUS 169. Again, from (3) and (4), Art. 168, dsj \dsj \dsj ' \ds J \ds where s' denotes the length of the arc of the evolute measured from a fixed point. Hence, ds ds' ds\ V Hence, s' ± p = a. constant, (1) since, if a derivative is always zero, the function can neither increase nor decrease, but is constant. It follows from (1) that A (s' ±p) = 0, As' = ± Ap. That is, the difference between any two radii of curvature PiC^, PsCs, is equal to the corresponding included arc of the evolute C1C3. 170. From the two properties of Arts. 168 and 169, it follows that the involute AB may be described by the end of a string unwound from the evolute UK. From this property the word evolute is derived. It will be noticed that a curve has only one evolute, but an infinite number of involutes, as may be seen by varying the length of the string which is unwound. EXAMPLES 1. Find the coordinates of the centre of curvature of the cubical parabola (Art. 130), a==j, = a;^ a' + 15x* „ a*x-9x' Ans. a = — 5 , p = 6 arx 2 a* 2. Find the coordinates of centre of curvature of the semicubical parabola (Art. 130), ay^ = a^. Ans. « = — If, ^ = ^(- + |)Vl- CURVATURE. RADIUS OF CURVATURE 205 3. Find the coordinates of the centre of curvature of the catenary (Art. 128),2, = |(e-«+e""). „ , 4. Show that in the parabola (Art. 129), as + j/^ = a', we have the relation u + p = 3{x + y). 5. Find the coordinates of the centre of curvature, and the equa- tion of the evolute, of the hypocycloid (Art. 132), x' + y'^ — a^. Ans. a=a +3x^ y^, j3 = y + 3x^y^, (a + ^)*+(«-j8)*=2a*. 6. Given the equation of the equilateral hyperbola 2xy = a!', show that a + l3= ^y+/^\ «-j8 = ^^^'. or a'' Thence derive the equation of the evolute, (a+;8)*-(a-|8)^=2ai 7. Find the equation of the evolute of the cissoid (Art. 125), rfij= ^ . Ans. 4096 a'a -I- 1162 a^/32 + 27 j8* = 0. 2a — X CHAPTER XVIII ORDER OF CONTACT. OSCULATING CIRCLE 171. Order of Contact. Let us consider two curves whose equa- tions are y = 4,{x) and y:=\a) = \jj'(a), the curves have a common tangent at P. The curves are then said to have a contact of thej^rs^ order. If besides, for x = a, the values of —^ are the same for both curves, that is, if <^ (o) = ^t (a), ^'(a) = !/-'(«), and ^"(a) = i/-"(a), the curves have contact of the second order. In general, the conditions for a contact of the nth order at the point X = a, are ,^(a) = V(«)' <^'(«) =•/''(«). <^"(«) = '/'"(«)> -> riO') = r(a), and ^"+^(a)^i/'"+'(a). 206 ORDER OF CONTACT. OSCULATING CIRCLE 207 In other words, for x = a, „, dy cPy da;"' dx' daf' must all have the same values, respectively, taken from the equations of both curves ; and " ' dx"+^ must have different values. 172. When the Order of Contact is Even, the Curves cross at the Point of Contact ; but when the Order is Odd, they do not cross. Let us dis- tinguish the ordinates of the two curves by Y=(x), and y = \li(x). In the figures Y refers to the full curve, and y to the dotted curve. If Y—y has the same sign on both sides of P, as in the first figure, the curves do not cross at P; but if Y—y is positive on one side of P and negative on the other, the curves do cross at P. Let OM=a, MM^^h. Then P^Q^ = Y-y=(a + 7i) -il/(a + h). 5^^ p ^-=^^— _ .yA Q-^ ^.yf f/ / f // Q= ^ / / Ma M M, Ms M Ml X Expanding by Taylor's Theorem, P,q, = i>{a) + li"'(a)-r'(a)] + ' (2) For sufB.ciently small values of h the sign of the lowest power de- termines that of the second member, and hence the sign of PiQi will remain unchanged when — h is substituted for h, giving P2Q2, as in the first figure. Thus when the contact is of the first order, the curves do not cross at the point of contact. Again, suppose the contact of the second order ; then PiQ.= "{a) = i'"{a), and (2) becomes — (a^ — b^) sin , where (j> is the eccentric angle. Ans. (ax)i + (by)^ = (a' - b^)i 6. Find the envelope of the straight lines represented by a; cos 3 + y sin 3 e = a(cos 2 0)^, 6 being the variable parameter. Ans. (of + y^'^ = a^ {af — y^), the lemniscate. 7. Find the envelope of the series of ellipses, whose axes coincide and whose area is constant. The equation of the ellipses is -,+f-!=i (1) a^ ¥ a and 6 being variable parameters, subject to the condition ab = 'k-, (2) calling the constant area rf. Substituting in (1) the value of b from (2), ^^ «y^ (3) in which a is the only variable parameter. Differentiating (3) with reference to a, we have a? ¥ ^ ' Eliminating a between (3) and (4), we have 4 x'y- = k\ 220 DIFFERENTIAL CALCULUS Second Solution. Differentiate (1), regarding both a and b as variable. a? da ,ifdb ^ /-, Differentiating (2) also, we have hda+adb = () . (6) From (5) and (6), we have ar y- «- (7) From (7) and (1), Substituting (8) in (2), 4 x-y- = /t*. 8. Find the envelope of the circles whose diameters are the double ordinates of the parabola if — 4 ax. Ans. y^ = Aa{a + .v). 9. Find the envelope of the straight lines - + ^ = 1, a b when a" + b" = fc". „ 10. Find the envelope of the ellipses — + 1/^=1, cv' h- when a + 6 = &. Ans. x^ -\- y^ = k^. 11. Find the envelope of the circles passing through the origin, whose centres are on the parabola 2/^ = 4 ax. Ans. (cc + 2a)2/^ + 3?* = 0. ENVELOPES 221 12. Find the envelope of circles described on the central radii of an ellipse as diameters, the equation of the ellipse being a- b^ 13. Find the envelope of the ellipses whose axes coincide, and such that the distance between the extremities of the major and minor axes is constant and equal to fc. Ans, A square whose sides are {x ± yY = Jc^- IIN^TEGRAL CALCULUS CHAPTER XX INTEGRATION. STANDARD FORMS 184. Definition of Integration. The operation inverse to differ- entiation is called integration. By differentiation we find the dif- ferential of a given function, and by integration we find the function corresponding to a given differential. This function is called the integral of the differential. For instance, since 2xdx is the differential of a^, therefore x' is the integral of 2xdx. The symbol | is used to denote the integral of the expression following it. Thus the foregoing relations would be written. d (ay') = 2xdx, | 2xdx = x^. It is evidently the same thing, whether we consider this integral as the function whose differential is 2xdx, or the function whose derivative is 2x. As regards notation, however, it is customary to' write I 2xdx = a^, and not j 2a; = x". 223 224 INTEGRAL CALCULUS In other words, /d is the inverse of d, and not of — dx Thus the general definition of j (x)dx is that function whose differential is ^(x)dx; the symbol | denoting "the function whose differential is," in the same way that the inverse symbol, tan'"^ denotes "the angle whose tangent is." Integration is not like differentiation a direct operation, but con- sists in recognizing the given expression as the differential of a known function, or in reducing it to a form where such recognition is possible. 185. Elementary Principles. (a) It is evident that we may write j 2 a; da; = a;^ -)- 2, or | 2 a; da; = a^ — 5, as well as \2xdx = 3?; since the differential of a? + 2, as well as of a;^ — 5 is 2 a; da;. In general | 2 a; da; = a;^ + C, where denotes an arbitrary constant called the constant of integra- tion. Every integral in its most general form includes this term, + C. (h) Since d{u ±v ± w) = du ±dv ± div, it follows that j {du ±dv ± dw) = I du ± i dv± j dw. INTEGRATION. STANDARD FORMS 225 That is, we integrate a polynomial by integrating the separate terms, and retaining the signs. (c) Since d{au) = adu, it follows that i adu = a I du. That is, a constant factor may be transferred from one side of the symbol | to the other, without affecting the integral. 186. Fundamental Integrals. Since integration is the inverse of differentiation, to integrate any given function we must reduce it to one or more of the differentials of the elementary functions, ex- pressed by the fundamental formulae of the Differential Calculus. Corresponding to these formulae we may write a list of integrals, which may be regarded as fundamental, and to which all integrals should, if possible, be ultimately reduced. AVe shall then consider in this chapter such examples as are integrable by these formulae, either directly, or after some simple transformation. du- I. Cw^.. J n + 1 II. r*^=iogM. mJ u III. {a"du-- "• /a»c log a IV. Ce^du = e". V. j cos udu = sin u. VI. j sin udu = — cos u. 226 INTEGRAL CALCULUS VII. I sec^ u du = tan u. VIII. j cosec^ M dw = — cot u. IX. I sec u tan M dit = sec M. X. I cosec u cot udu = — cosec u. XI. Ttan M dw = log sec It. =r — -Ce^.^^^*- ^ XII. fcotwdit^logsinM. ■=: — -c*-^ ^*-e '^ XIII. j sec udu = log (sec m + tan u) = log tan ( - + ^ )• cosec M dw = log (cosec u — cot u) = log tan - • XV. r^*^ = i tan-' ^, or = - i cot-' ^ ■ J u' + a^ a a a a XVI. -^ - = --log— — , or = — -log— —. J w —or 2a u + a 2a a + u XVII. r '^" = sin-' - , or = - cos-' - • XVIII. f i!!_ =log(w + V^?T^). XIX. I =-sec '-, or = cosec '-• J u-Vu^ -a' a a a a -ir-tr r du XX. I — — = vers d(t _i 11 — = vers - • mTEGRA.TION. STANDARD FORMS 227 INTEGRALS BY I. AND II. 187. Proof of I. and II. To derive I., since d(M"+^) = (n + 1)m" du, therefore tt"+^ = f{n + T)^ du = (n + 1) Cu" du, by (c), Art. 185. Hence I u" du = /•• Formula II. follows directly from J 1 du d log u = — u It is to be noticed that I. applies to all values of n except w = — 1. For this value it gives ; = — =00 . J Formula II. provides for this failing case of I. EXAMPLES Integrate the following expressions : 1. ( x*dx. f If we apply I., calling u = x, and w=4; then du=dx. Then we have I ai*dx = — \- C, adding the constant of integration C, according to ./ 5 (a), Art. 185. 228 INTEGRAL CALCULUS 2. C(x^ + l)h'dx. If we apply I., calling u = oe' +1, and « = - ; then du=2 xdx. We must then introduce a factor 2 before the xdx, and conse- quently its reciprocal - on the left of | C(ay' + l)^x dx=^C{x' + lf2x dx, by (c), Art. 186. 2 3 3 ■ 2 o r (a;^ - a^)c?a; _ 1 r (3x^-3a^)dx J ct^-Sa'x^sJ a^-Sa'x = llog (a^ - 3 a'x) = log (a^ - 3 a'x)^ + C. o By introducing the factor 3, we make the numerator the differ- ential of the denominator, and then apply II. 4. f(2a^-3x'+12a?-3)dx = — - ^'-i-3a;^ -3a;+ C. (:^ - 2yx' dx = ^-'^ + i^- 2 x' + a 7. C (x^ - 2yx dx = ^^ ~ ^)' + C INTEGRATION. STANDARD FORMS 229 11. r(a;i + a;-iYda; = ^ +1^+6 a; + 12 x^- 3 a;- J + 0. 12. r(^^llZd2/=.^_3|! + 3 2/_iog2/+(7. J y 6 Z 13 p + l)(x-2)^^^3^_3g!_6^i^ J ^* 7 4 14. r(af' + l)'a^da; = ^^5^±i^ + C. 15. r(aa;2+6)*a;c^a;. 16. C(ax + b)'dx. 17. r(aa^ + &)Vd.<;. 18. fiax" + byx''-''dx. 21. /(^"'-l)"'^- 22. J(a+logO 230 INTEGRAL CALCULUS 23 / e' + af -dx. 24. J'(a-logioa;) dx X 25. a 1 +^^ e^ da;. 26. Tsin^^ cos 6 dd. 27. C(e'^ + sin 261) (e^^ + cos 26l)d0. 28, ( tan^ X sec' X dx. 29. | sec^aj tan a; cZa;. 30. r(sin'"6l + cos"^) sin e cos 6 dd. 31. r(sec e + tan e/" secede. 32. r(sin + cos <^)"(sin2 - cos^ (^)d(^. 33. ("(a* + 6)" a"^ da;. 34. /sin~^ X dx 35 ■J, dx (1 +a;2) tan-'a; A rational fraction, whose denominator is of the first degree, may be integrated directly or after being reduced to a mixed quantity. 36. r-^ = flog(4a;-3)+(7. J 4:X — 3 4 37. J 2.»; + 3 .« = ~-^log(2a; + 3)+0. ^x^-lx' + l. _a? 2,3? dx 1 r 2o^-lx'- ] J 2a;-l ■dx 2 4 log(2a;-l) + a INTEGRATION. STANDARD FORMS 231 J bx + a b b'- ^y ^ J^ iO. /0(j I fj, Cu CtiSu — — — dx = ~-\- —- ~\- a^x + o? log (x — a)-\- C. X — Ct O Jj 41 • J x + b 2 42 J X— a 3 INTEGRALS BY III. AND IV. 188. Proof of III. and IV. These are evidently obtained directly from the corresponding formulae of differentiation. EXAMPLES ^ 4 6 log a 2 log 6 2. r(e'" + e-'^'fdx -- + 3e'"-3e- + C. 5x 4x 7x >:^D!a.=3«! 4. riiL^i!:^da, = ^i^-6e3^ 3e- C. 232 INTEGRAL CALCULUS 5. j (e*+'' — 6'™+') dx. 6. j (e"""'cosa; — a'='"^sin2a;)f?«. 7. Cfe''^x + -\dx. 8. fie^" ^ sea d-e'^Hand) sec Ode. J e^ J J ^ ^ log(a&') /„mx-\-nl.^x-\-q a""+"6'"+« da; = — + C. log {arh'') J e^ e"^21oga-l 21og6-iy J^ ' 21og6 log(a6) 21oga INTEGRALS BY V.— XIV. 189. Proof of V. — XIV. It is evident that V. — X. are obtained directly from the corresponding formulae of differentiation. To derive XI. and XII., /, , C— sin udu , 1 tan udu = — I = — log cos u = log sec u. J cos 14 /J. J /'cos udu ^ cot udu= \ — = log sin M. J sinit To derive XIII. and XIV., /, /'sec w (tan M+ sect*) d?t /"secMtanttdw + sec^Mdi( secMdM= I 5^ ■ '- — = I ■ ' J see u + tan u J sec u + tan u = log (sec u + tan u). /, /"cosec M ( — cot u + cosec u) du cosec udu= I !^ ■ — ; ^ — J cosec u — cot u = log (cosec M — cot m). INTEGRATION. STANDARD FORMS 233 By Trigonometry, 2siii2- , 1 — cos M 2 , u cosec M — cot u = — : = = tan - • 2 sin -cos - 2 2 If we substitute in this, ^ + m for u, we have secM + tanw = tan( - + ^ 4 2y Thus we obtain the second forms of XIII. and XIV. ., r/ • o , a ■ x\j COS 3 X , sin 5 x 1. I ( sm 3 a; + cos 5 a; — sin - da; = J V ^J 3 5 + 2cos| + a n Cf ■ x-\- a , x-\-'b\ ^ x-\-a , ■ xA-h , r^ 2. I sm — h cos — ■ — ]dx= —m cos — h m sin — ■ 1- C. J \ m n J m n I 3. I — ■ — ax = — (tan mx + sec mx) + G. J cos ma; m 4. I (sec 6 a; — tan 5 a;) sec 6 a; c?x. 5. j (sec 2 ^ + tan 2 6) dS. 6. Cf^ + ^^\ae. 7. r(sine-yerse)^de. JVcos'^ sin^y J^ ' 8. I dx = cos a; — 2 log (1 + cos a;) + O. J sm X 9. Cl^dx. 10. C^^dx. J cos'^a; J sm^a; ^^ r Bec4>d = l]og(atan<^ + 6) + a J a sm y Ax, and AA < (y + Ay) Ax, AA. Ax dA ■>y, ■, AA ^ , . and -^ (x) Hence fM = A + (x-a)^- {x) '^{x) This being an identical equation is true for all values of x. 252 INTEGRAL CALCULUS If we put a; = a, we have A = ^-f^, since by hypothesis ^ (») does not vanish when x = a. "^ ^'^' Thus we have the following rule : To find A, the numerator of the partial fraction , put x = a X — a in the given fraction, omitting the factor x — a itself. For example, having written equation (1), we find A by substitut- ing a; = 2 in the given fraction — — '- , omitting the factor o mu- • (x-2)(x + 2)x X — 2. This gives ^ ^/v t^ y 4(2) To find B, substitute x = — 2, omitting the factor x + 2. 4-12-8 ^ ^ -4(-2) To find C, substitute x = 0, omitting the factor x. (7=51 = 2. EXAMPLES The constant of integration C will be omitted in the examples in this chapter, and the following chapters on the integration of func- tions. , r x*dx a? , 3x\ „ ,1 (x-2)w 1. I = — h'x + logi i— . Jx'-3x + 2 3 2 ^ ° x-1 2 r (x'+x+l)dx ^ 1 , (x + l)(x-3y^ ' J x'-4:x' + x + 6 12 ^ {x-2)^ ■ 3 r(iv +iydw ^l^^ J2ic + l)(2w -ly INTEGRATION OF RATIONAL FRACTIONS 253 4. r "^ -ho- (^+^)° J {x + l){x +3){x + 5) 8 ^ (a; + l)(a; + 6)^" 5 r ^Jh ^1^ (3x-m3x-2f ' J (2x-l){3x-l)(3x-2) 18 ^ (2a;-l)« ^ fr-^^l^Tr^— ^'^''=-T^M(bx -a) +— ^log (ax-b). J{ax — b)(bx — a) ab—h' , ab — a^ ^ r (x + ay-ch- __l^^ J2x + a)(x-a)\ J 2 x? — cuy^ — a'x 6 a;* o r(x + a + bYdx , 6^ , / , \ , aP ^ / , j,\ J (x + a)(x + b) b — a a — b 9 c '^y = 1 ioc Cy+")'(y-^)''"' ■ ^(2/ + «)(2/'-l) 2(^^-1) ^ (2/ + l)»+^ ■ jQ / -(ffl^-6^(a^ + l)c?a; ^ 1 ^ (ax + b){bx - a) ■ J (a^j;^ - ¥) (bV - a') 2ab ^ (ax -b)(bx + a)' „ r (x + l)dx _ 1 . (x + 5r(x-7r ' J (a?- 19)^- 4 (x + 8)^ 360 ^ (« + S)\x - If ' 12 C ^^ + -'-)^^' ■ J 4:3^-17 x^ + 4-x = ^log(^=^-^log[(2x + l)(2x-iy] + hogx. 254 INTEGRAL CALCULUS 198. Case II. Factors of the denominator all of the first degree, and some repeated. Here the method of decomposition of Case I. requires modifica- tion. Suppose, for example, we have L -^ + 1 d.. x(x - If If we follow the method of the preceding case, we should write ai> + i ^A B C D x{x — Vf X x — 1 x—1 x — 1 But since the common denominator of the fractions in the second member of this equation is x(x — 1), their sum cannot be equal to the given fraction with the denominator x(x — Vf. To meet this objec- tion, we assume 3^ + 1 ^A BO D x{x~lf X {x-lf {x-lf a;-l Clearing of fractions, a? + l = A{x-Vf + Bx+ Cx(x - 1) + Dx{x - 1)^ = (A + D)3? + (- 3 .4 + C - 2 D)x^ + (3A+B~C+I))x- A. Hence A + D = l, (1) = 3A+C-2D = 0, (2) 3A + B-G+D=0, -A = l. Whence ^= -1, B = 2, C= 1, Z> = 2. Therefore i^ + ^= -1+ 2^^ + ^ x{x-iy X {x-iy (x-iy x-i INTEGRATION OF RATIONAL FRACTIONS 255 = __^ +log(^:=iI^ {x—iy X The nnmerators A and B may be determined by the short method given for Case I., and then G and D may be found by (1) and (2). EXAM PLES a;^— a^ + 1,. x^ , , 1 ■, 1 , ■, a;— 1 2 r a?dx ^ 2-Zx 1 ^ x-1 ' J {x + l){x-lf 4(0;-!)=' 8 ^ a; + 1 3 C ^'^^ = _ 1 + - lo ^'' ■ Ja;(x^-4)^~ CB^_4 4 °^x^-4:' X r (19a;-32)c;a; ^ 1 3 ^^ 2 a; - 3 J (4a; + l)(2a;-3/ 4(2a;-3) 4 °^4a; + l' J (9 a!= - 4)2 18(9 a;^ - 4) 216 ® 3 a; + 2 n r x^ + a , a- — 3 aa; , 1 xF 6. rr-^2<^^ = ^ +^°s ./ (a/"^ — axy x' — ax x — a 7. rf--±lY,.= . + l|2^— 24 +81og(.-l). J \x—lj 3(x — iy x — 1 256 INTEGRAL CALCULUS Q r xdx ^ _ 2a;-l 1 ^ a; - 2 + V3 10. f C^ + c^ + Sf d,^ l_f_^ + ^ J (a!+a)^(a; + &)2 {a-by\x + a x + b (a — bf (a — by 199. Case III. Denominator containing Factors of the Second Degree, but none repeated. The form of decompositiou will appear from the following example, J xiv? + 4) 1X7 o X -\-Vl A , Bx ^ C .-I s We assume ,.,,, =-+ , 7/i ' W and in general for every partial fraction in this case, whose denomi- nator is of the second degree, we must assume a numerator of the form Bx + G. Clearing (1) of fractions, 5 .x + 12 = (^ + B)a;^ + Cx + 4 A .1 + 5 = 0, C=6, 4 J. = 12. Whence .1 = 3, J3 = -3, (7=5; 5x + 12 3,-3a; + 5 therefore a;(a!^ +4) x a^ + 4 J a-= + 4 J a;^ + 4 J a;^ - + 4 J a;2 + 4 J a;^ + 4 -|log(a;= + 4) + ^tan-i|. INTEGRATION OF RATIONAL FRACTIONS 257 Hence C^;^±^ dx - 3 los "" + - tan"^^ • Take for another example, r (2x'-3x-3)dx J (x~l){x'-'-2x + 5)' This fraction is decomposed as follows : 2a;2-3a;-3 1 3a;-2 {x-l)(a^-2x + 5) x-1 a?-2x + 5 r (3 a; - 2)dx ^ /- (3a;-3)da! r tfa; J a;2-2a; + 5 J a;^-2a; + 5 J a^-2 a; + 5 = |log(a;^-2x + 5) + itan-i?^. / - (2a;^-3a;-3)fe ^ ^ {x'~2x+5)i 1 ^^^-i^-l J (x-l){3?-2x + 5) ^ x-1 2 2 The integration of any fraction with a quadratic denominator like the preceding, ( >' ^~ — —^, may be shown as follows: Joi? — 2x + 5 Having written the denominator in the form (x + cCf + 6^, we have r {px-\-q)dx _ r p{x + a)dx C (q—pa)d'. _ {x + ay + b'~J {x + ay + b'^J {x + ay + X = 1 log lix + af + h'^l^-^-^ tan-^. 258 INTEGRAL CALCULUS EXAMPLES 1 r32afi+3., 8a? „ ,1, x^ ,0/54.-1 2a; 1. I — - — ' — dx= 6a; + -loB — h3v3tan' — :;■ J 4.afi + 3x 3 2 ^4x^ + 3 V3 • J aA+a^-2 6 ® (x + lf 3 V2 - r x'dx a? , 1 T a; - 1 , 1 , _, 4. H — T = ^ + I^°S—-r + ^ tan-la;. J a;' — 1 3 4 a; + l 2 When tlie given fraction and the denominators of the partial fractions contain only even powers of x, they may be regarded as functions of x', and we may assume A, B, C, etc., as the niimerators of the partial fraction. In the following example, the partial fractions may be assumed as X- + a^ X- + b- J {x^ + a^){x- + b') a' — b-\ b a 6. C (/-y =Ytan-i2x-tan-'g;Vltan-i- ^'^ J (4a;= + l)(a;= + 4) 6\ 2j 6 2 + 2 ar ♦ J (aV + b^{bV + a') ab ab(l - a? r {x-l)dx ^li •'^-Ga^ + lS Atan-i-:"_:z. J x(3?-Qx+13) 26 ° x^ 13 2 INTEGRATION OF RATIONAL FRACTIONS 259 g r (3a?-2x-20)dx ^1^^ (2 a;^ - 6 a; + 5)' + A. tan-i^- - - tan-i (2x- 3). J f-1 6 2/- + 2/ + 1 V3 V3 r(..^ + 2)rf.. ^ 1 ^ x-^ + X + 1 V3 ^^^_. xV3 . Ja;< + ar' + l 4 °a;^-a; + l 2 1 - a^ 12. r»;l!^^J_i ^--^»V2+l ^J^tan-'i^. Jw^ + 1 4-^/2 ,,,24-»iVJ-l-1 2 a/2 1-m)^ ' + 1 •4V2 w' + wVJ+l 2V2 1-")' 13. r ^^^ = __l_ + llog Ja^_ar^ + a^_l 6(a;+l) 4 ^ ^'^^ - _l_^li..=L=:l ^taii-2^^. a^_ar^ + a^_l 6 (a; + 1) 4 °a;,+ l 3v'3 V3 200. Case IV. Denominator containing Factors of the Second De- gree, some of which are repeated. This case is related to Case III., as Case II. to Case I., and requires a similar modification of the partial fractions. Tor illustration take / 2^ + »'" + 3dx. We assume 2a?' + a!^ + 3 ^ -4a; + -B Cx + D (a^ + 1/ (a^+1)^ x' + l ' 2x^ + x' + 3 = C3? + Dx'+(A + C)x + B + D. A=-2, B = 2, C=2, D = l. 260 INTEGRAL CALCULUS Therefore ^^ + ^ + ^ = -2«-- + 2 + l^+l (x' + lf {a? + lf x' + l r -2x + 2 ^^^ r 2xdx ^r dx J (x' + If J (a;- + 1)^ J (af + iy = _J__ + o f dx X- + 1 "^ V Or + 1/' To integrate the last fraction, we use the following formula of reduction, (a^+ay ~2(n- 1) a-' l_(a.-^ + a^-' '^ ^-^ '^ ~ )j(^fj^^f^ij /dx -— — by making it (ar+a )" depend upon I -^ • By successive applications the given ^ ^ J (x' + d'Y-^ r dx integral is made to depend ultimately upon | -, which is -tan '-• a a * This formula may be derived as follows : -^r ^ 'l = A[x(x2 + o2)-«j = (a:2 4-a2)-»-2na;2(a;2 + a2)-»-i dx\_{x'^+ a'^)"-} dx = (a;2 + a2) -" - 2 ii [(a;2 + a^) - a^j (a;2 + a^)-"-! = (1 - 2 n)(x2 + a2)-n _^_ 2 ma2 (x^ + a^)-"-'-. Integrating both members after multiplying by dx, 2 = (l-2n) C — + 2na^C , '^^, ;. Substituting for )i, re — 1, we have 2 fn - 1) a2 f — = ^ + (2 re - 3) f -^ INTEGRATION OF RATIONAL FRACTIONS Substituting in the formula w = 2 and a? = 1, we have c_§_x _ir ^^ , r (to 1 X , i\ _i 261 whence and (a;2 + 1)2 2 r -2x+2 J (af + iy dx = -r^ +-^^ + ta,n-^x, x^ + 1 a? + l '2 a:^ + a^ + 3 (-2^ dx = x+1 {x' + iy- -- a^ + l A partial fraction of the form P^ + 1 + 2 tan-i x + log (a^ + 1). , by substituting x + a=z, becomes ^^f ^ ^^ ' *^® integration of which has already- been explained. v^ + " J For example, if a; — 3 = 2, I — ^ ^^—^ =1 — ^ ^ ' 'J (ar^-6a; + 12)= J (z^ 5 "4(2^+3)= (2^ + 3)' dz ■dz. + 16 f— (2' + 3)» By the formula of reduction, /, dz (f+sy 12 liz' + sy J (z' + syj 1 1 r z r dz " I 4'6|_z^ + 3 Jz2 + 3j = ^^ h 12 (z^ + Sy 4: I J 4- —I ^ 12 {z' + Sy ^ 24 (2^ + 3) ^ 24 V3 ^'^ VS' Hence r 5g + 16 ^_ 16Z-15 J (z' + Sy 12(2^ + 3)^ 22 2 4. -1 2 tan^ — (2^ + 3)^ - 12 (2^ + 3)^ ' 3 (2^ + 3) 3 V3 V3 (5a; + 2)c?.r _ 16 a;- 63 (a;^-6a;+12j'= 12{a^-6x + 12y I 2 (a; -3) , 2 3(a:^-6u; + 12) 3V3 tan _iX--3 V3 ■ 262 INTEGRAL CALCULUS EXAMPLES -—i— -dx = -^ h- log (or 4-1) + -tan ^x. ^•/(5^^^^=^+^^-^«*^"-^^ 2a^a o„j. \X f. dx = - ' — — H = tan" (4x^ + 3)= 8(4a;=+3)^ 16 V3 V3 For the following example, see note preceding Ex. 5, Case III. 4. a? -\-x (4x2 + 9/(9a;^ + 4)^ 2(4a;2+ 9)(9a;^ + 4) , 1 , _i 13a; A tan ' • 156 Q-&a? 5 r ^ + ^x-21 ^^^ 3(a; + 7) ■ J (x= + 4a; + 9)2 2(a;^+4a; + 9) + Jlog(a,'^4 4a; + 9)-^tan-'^. 6 f (g^ — 2)da; ^ a; + 4 • J (»2 + a; + l)(a!' + x + 2)^ 7(a!2 + a; + 2) + _12_^^_,2^ + 1_^^^^_,2.^_+1. 7V7 V7 V3 V3 /•_9x^^^_^^ Ij J«+l)l+V3tan-2^-l^ - f dx _ 12ai^ + 36 a! + 29 [(a; + 2)^ -(a; + 1/7 2(2a; + 3)(2a;2 + 6a; + 5) -3 tan-' (2 a; + 3). CHAPTER XXIII INTEGRATION OF IRRATIONAL FUNCTIONS 201. We have shown in the preceding chapter that the integral of any rational function can be expressed in terms of algebraic, loga- rithmic, and inverse-trigonometric functions. We shall now consider the integration of irrational functions. 202. Integration by Rationalization. Some integrals involving radicals may be integrated, by reducing them to rational integrals by a change of variable. This is possible, however, in only a very limited number of cases. This process is sometimes called integra- tion by rationalization. p 203. Integrals containing (ax + by . Such an integral may be rationalized by the substitution ax + b = »'. For example, take | - x^dx (2x + sy Assume 2a5 + 3 = »', ^—~~3 — ■• <^^= ^ • Then +9 ^)= |5(|' - f^ + 9)= ^-| (2 a=+3)^(8 a;^-18 a;+81). 3/'/_3/ , „.\ 2,z(^ \\l 2 263 264 INTEGRAL CALCULUS Another example is r_xdx_ _ •^ Vx + l Assume x = z^, dx = 2z dz. Then r_£d^^ r2^ = 2 C(z^-z + l- J-)dz = 2r5! - ^ + a - log (2 + 1)1 = x^ - a; + 2 K* - log («* + 1)1 £ r 204. Integrals containing (»« + &)', (ax + h)', ••■ . In this case the integral is rationalized by the substitution ax + b = z", where n is the least common multiple of q, s, •••, the denominators of the fractional exponents. Take, for example, j ~ . ■^ (« - 2)i +{x- 2)i Assume x — 2 = !^, da; = 6 g" dz, (x-2)i = :^, (x-2)^ = z\ /^ / -I \ dz r dx ^ r fiz'dz ^g rz^dz ^g r/ -^ 1 ^ (a;_2)* + (a;-2)* -'2' + 2' Ja + l J I, z + 1 = 6r|'-» + log(« + l)1=3(a;-2)^-6(a;-2)* + 61og[(a;-2)* + l] EXAMPLES 1. f '",JlL- da; = 2Va;-2 + V2tan-\/i^ -' x^x-2 \ 2 2. p^A^rrsdx^i^^f^ ' (4 a; -3)^ 4a; -3 9' 26 3 10 / l + ^^^^ ^i^^''~^^'~^^''~^^^+^°^^^ + ^^'"~^^- INTEGRATION OF IRRATIONAL FUNCTIONS 265 . rx^ + a;^ + 1 , 12 a;" , „ i a i , A^ /■ i , -i\ 4. I — ^ -' — dx = l-2a;^ — 4x* + 41og(x^ + l). ^ ic' + a;" 7 5 r fdy _ 4.if-&f-&y-l ' ^ (42/ + l)« 24(42/ + l)i 6 r %==—=^ +21og(V2¥:^ + l). •^w + V2w — 1 v2w — 1 + 1 7. /^ = 2.^-3^V3|:^-|log(2.A + i). •^ i^ + 2f^ ^ ^ 4 8. ra^Va»T& c?a; = ?%^±-^ (15 a=a;2- 12 a6a; + 8 6=). J " 106 a' g_ r dx = 2(3 x + l)i - 4 tan-^^+lI* . •^ (3x + l)* + 4(3a! + l)^ 2 10. fVa + l-l ^^ ^ 2 VSTl - log (a; + 3) - 2 V2 tan-^ J?-±i 11. r (^^2}^ = 3(2.,_3)* + llog2^^)i±3 '^ (2a;-3)* + 6a;-9 ^ 8 (2a;-3)^ 3V3, ,{2x-^ , 4 V3 jg r dx -^ -^ V2a; + 1+V^^^ = 2V2^n - 2V^^^ + 2V3 Aan-UlEl _ tan-'-Jl^±l = 2V2^+1 -2V^3l + V^faos-^^^^-cos-^'^—^. \ x + 2 X + 2J 266 INTEGRAL CALCULUS 13. C—^^ = 2 a;i+ hog (2 xi - 1) + ^ log (x* + 2) (2a;i-l)(a!*+2) 16 V2, _, »' ^— tan ' — 7= • 9 V2 14. r (^ + 2)dx ^2V^+l + |log(a; + 2-Va; + l) •^ (x + l)t + l 3 -^ log (1 + ^+1). O 205. Roots of Polynomials of Higher Degrees. — In the rationalizar tion of irrational integrals we now pass from roots of binomials of the first degree to roots of polynomials of higher degrees. Here rationalization is limited to the square root of an expression of the second degree. 206. Integrals containing Va^ + ax + b. This may be rational- ized by the substitution For example, consider | Va;^ + ax + b = z — x. dx X -\/x^ — x + 2 If, following the method of the preceding articles, we assume Va^ — x + 2 = z, ct? — x + 2=z^, the expression for x, and consequently that for dx, in terms of z, will involve radicals. This diflB.culty is avoided by assuming ^a? — x + 2 = z — x, —x + 2 = z' — 2zx, cancelling x^ in both members. z- 2. ^^2(z^-z + 2)dz_ X — - r, dx 22-1' (22-1) Va^-a; + 2 = g-a; = ^'~^ + 2 . 22 — 1 INTEGRATION OF IRRATIONAL FUNCTIONS 267 Hence, 2{z'-z + 2)dz .a-V2 X -Z(z' — z + -J,)az {2z-iy ^ r2dz ^ J. ^-2_z^-z + 2 Jz^-2 V 22-1 2z-l = -— log V2 Z+V2 Substituting z = Va^ — x + 2 +x, dx _ 1 , Vl^^^^+2 + x — ^/2 C ^ = J_iogX| Va;^-a; + 2 V2 Va^-a; + 2 + a; + V2 207. Integrals containing ■\/ — ay' + ax + b. This may be rational- ized by the substitution V— ay' + ax-^ b = V(a — x)(fi + x) = {a — x)z or =(fi + x)z, where a — x and p + x are the factors of — a? + ax + h. These factors will be real, unless ^—x^ + ax + h is imaginary for all values of x. dx Take, for example, | — J X V2 + a;-a^ Assume -\/2 + x-x' = V(2-x)(l + a;) = (2 - a;>. t i rn \ s 22^ — 1 , 62d2 14-a! = (2 — «)«', « = — ^ — ;-, d« = 22+1' (a» + l) 3« s V2 + a: — »^ = (2 — a;)2 = ^iXj' Therefore, -' a;V2 + a!-a^ •>' 2»2-l V2 zV2 + 1 Substituting 2 = \ g^^ ' r '^"^ =^ , V2 + 2a;-V2-iB J xV2 + x-x' V2 V2 + 2a; + V'2-a; 268 INTEGRAL CALCULUS EXAMPLES / - = tan ' — —^ — -^ x^x- + ix — 4: J ^x' + ix ^^^ 8 +\os:(x + 2+^x' + ix). = / dx (2ax-x'f a?-\/2ax-x^ r ^- ^^-^ = 2 tan- Jl^ - A tan- J^p^. ' _,x + l 2 _, ^cf = cos ^ — i- — • cos— V3 S-x rV^+^ ^^+a:_4a^ 6aj 3^-a •>' 3a; + 4a 6« 9^9 ^z + 3a' where z = x + Vx' + a?. 208. Integrable Cases. — The preceding articles include those forms of irrational integrals that can be rationalized. In general, integrals containing fractional powers of polynomials above the first degree — except the square root of polynomials of the second degree — cannot be rationalized, and cannot be integrated in terms of the elementary functions, that is, cannot be expressed in terms of alge- braic, exponential, logarithmic, trigonometric, or anti-trigonometric functions. INTEGRATION OF IRRATIONAL FUNCTIONS 269 Every integral may be regarded as defiuing a certain function. It has been shown in Art. 192 that if f{x) is any continuous func- tion of X, i f(x)dx is a function of x, which may be geometrically represented by an area bounded by the curve y =f(x) ; but this cannot always be expressed in terms of the elementary functions. CHAPTER XXIV TRIGONOMETRIC FORMS READILY INTEGRABLE 209. It is to be noticed that any power of a trigonometric func- tion may be integrated by Formula I., when accompanied by its differential. Thus, /. „ , sin"+ia; C „ ■ j cos"+'a; sin" X cos a; cZa; = , I cos" x sin xax = • , H + 1 J n+1 /x - 2 J tan"+'a; T ,„ a -, cot"+ia; tan" X sec* xdx = , I cot" x cosec' xdx= , n+1 J n+1 /. , sec"+'a; sec" X sec x tan xdx — , n+1' f , , cosec"+'a! cosec" X cosec a; cot xdx= n + 1 Having in mind these integrals, the student should readily under- stand the transformations in the following articles. 210. To find I sin" a; dx or I cos" x dx. When n is an odd posi- tive integer, we may integrate as in the following examples : I sin* a; da; = i sin* x sin a; c?x = | (1 — cos^ a;)^ sin a; dx = j (1 — 2 cos^ x + cos* a;) sin x dx = — cos x + 270 2 cos^ X COS* a; TRIGONOMETRIC FORMS READILY INTEGRABLE 271 Another example is jcos'2a;da!= j cos^ 2 a; cos 2 a; da; = - j (1 — sin^ 2 as) cos 2 a; 2 da; If ■ o sin^2a; = - sm 2x — 211. To find j sin" x cos" x dx. When either m or m is an odd positive integer, this form may be integrated in the same manner as in the preceding article. For example, I sin^ a; cos' a; dx = j sin* a; cos^ a; cos a; da; = j sin* a; (1 — sin^ a;)^ cos a; da; // ■ 4 o • 6 , • g \ J sin'a; 2sin^a; , sin' a; (sin* a; — 2 sin' x + sm' a;)cos a; da; = — — 1 5 7 9 Another example is j sin' X cos^ xdx= i cos' a; sin^ a: sin a; da; = j cos' a; (1 — cos^ a;) sin x dx = r(costa;-cos^a;)sina;da; = -2^ + 29osi^. J 5 9 EXAMPLES 1 r ■ 7 J , ■i S cos* X , cos' X 1 . I sin' xdx = — cos X + cos'' x — 1 J o 7 o /^ q J • 4 sin'' a; , 6 sin' a; 4 sin' a; , sin' a; 2. I cos' a; da; = sm a; 1 1 — J 3 5 7 9 o r ■ B^ J o a;, 4 oa;2 ,x 3. I sin° - da; = — 2 cos - + - cos'* 7: — -z cos* - • J 2 ■ 2 3 2 5 2 A C ■ s± rj j-i sin'd 3sin" 2 = sin^ = /'*?J^ + tan ^ log sin <^ - ?!5^ - "^ 9. r^2£^da; = ?V3^^floga;-2Ui:^tan-H/ 10 r log(^ + 2) ^^^_log(^ + 2) ^^+1. 11. I tan~' - da; = a; tan~' -log(a;^ + a^)- J a a 2 12. ( a;^ tan~' xdx = — tan~^ a; — — + - log {ay' + 1). ^ 3 6 6 13. I sin~^ - da; = a; sin~^ - + s/a^ — 3?. J a a 14. r(3 «2 - 1) sin-i a; da; = (a;8 - a;) sin-^ x - (Izi^ 3a! — 2 282 INTEGRAL CALCULUS IK C ■ s J /cos' X \ , sin' X , 2 sin £c 15. \x3VD?xdx = xl cosa;)-! — 1 5— • 16. Cx (sec° X - tan« x)dx = x tan' « + ^ - ^-^^ + log sec x. 17_ flog (e- + 1) ^^ ^ ^ _ gl±l log (e. + i). 18. I log (a + Va^ + a^) da; = a; log (a + V»^ + a^) 4- a log (a; + ^a? + a^) — a;. In each of the following examples integration by parts must be applied successively. 19. pB'e-2*da; = -5^(4»' + 6a!''4-6a5 + 3). 20. C(e'^-x)Hx= — -^(4.x-l) + ^(2a?-2x + l)-^. 21 . fa;'-^ (log xfdx = ^" [(log xf - ^^^^ + ^1 • J n\_ n n^J 22. r«'sin2a!da; = ^^-^\in2x-/'^-^Vos2a;. 23 ra;(tan->a;)2da; = ^±l(tan-i x)^- a;tan-ix + ^ log {x' + 1) . 24. /af — a a; log {x + a) log (a; — a)dx = - — - — log (a; + a) log (a; — a) (x + aYi , , ^ {x~ of •. , s , ar" ■ N ^ ^- log(a; + a) - '' ^ log {x-a) + - • INTEGRATION BY PARTS. REDUCTION FORMULA 283 217. To find j e" sin nx dx, and Ce" cos nx dx. Integrating by parts, with u = e^, fe-" sin nxdx = - ^'" "°^ ^'^ + ^ fe-' cos nx dx. . . (1) J n nJ Integrating the same, with u = sin nx, fe'" sin na; dx = ^^^^5:^ -- fe""' cos na; da;. ... (2) J a aJ We see that (1) and (2) are two equations containing the two required integrals, I e" sin wa; da; and j e"" cos na; da;. Eliminating the latter, by multiplying (1) by n^, and (2) by a^, and adding, gives (a^ + n^ I e" sin nx dx = e'" (a sin nx — n cos nx) ; T, C ax ■ J e'"(asinwa; — ncosna;") /on hence I e"" sin na; da; = — ^^ i- (3) J a^ + n^ Substituting this in (1) and transposing, gives a C a^ J e*" (an sin na; + a^ cos nx) - I e'"cosma;da; = — ^ 1- nJ aP + n' ' ■, r oj: J e"" (n sin nx + a COS nx) ,., hence I e'°'cosna;da; = — ^^ — ■ '- (4) J a^ + n^ ^ ^ EXAMPLES The student is advised to apply the process of Art. 217 to Exs. 1-4. For the remaining examples he may substitute the values of o and n in (3) and (4). e*" sin 5 a; da; = —(3 sin 5 a; — 5 cos 5 a;), ^ cos 5 a; da; = — (5 sin 5 a; + 3 cos 5 x). 284 INTEGRAL CALCULUS 'J' J' e~^ sin xdx= — (2 sin x + cos a;), e~^ cos xdx= — (sin x — 2 cos a;). e" sin axdx = — (sin ax — cos ax). e 2 cos -da; = -7- ( 2sin- — 3cos- ). c /* sin 2 a; + cos 2 x j __ _ sin 2 x + 5 cos 2 a; 5./(e 6. I (e^ + sin 2 a;) (e" + cos a;) cZa: = — - + — (sin x + 2 cos a;) 1 o o , e"' / • o n o \ 2 cos' a; + — (sm 2 a; — 2 cos 2 a;) — . 5 3 e^ cos^ 3 a; da; = f- -^ (3 sin 6 a; + cos 6 x). j e' sir 8. \ e' sin 2 x sin 3 a; da; = — sin a; + cos x - 5 sin 5 a* + cos 5 x 13 I fx 9. I xe^ cos a; da; = — [5 a; (sin a; + 2 cos a;) — 4 sin a; — 3 cos a;], ^o 218. Reduction Formulae for Binomial Algebraic Integrals. These are formulae by which the integral, J a;'"(o + 6a;»)-''da;, may be made to depend upon a similar integral, with either m or p numerically diminished. There are four such formulae, as follows : INTEGRATION BY PARTS. REDUCTION FORMULA 285 Cx"(a + bx''ydx (np + m + l)b {np + m + r)bJ ^ ' ^ ' ix'^ia-ybvf)^ dx ^ ^^\a + b^y ^ npa T^. (^ + ^^y-i ^^^ .... (5) np + m + 1 np + m + lJ Cx"" (a + bx")" dx (m + l)a (m + l)a J ^ -^ J > ^ J ( af(a + bx")''dx n{p + l)a ^ nlj} + l)a J ^ ^ ^ ^ ^ Formulae (A) and (B) are used when the exponent to be reduced, m or p, is positive, (A) changing m into m — n, and (jB) changing p into p — 1. Formulae (C) and (J*) are used when the exponent to be reduced, m or p, is negative, ((7) changing m into m + n, and (D) changing p into p + 1. If, in the application of one of these formulae to a particular case, any denominator becomes zero, the formula is then inapplicable. For this reason, Formulae (A) and (B) fail, when np + m + 1 = 0. Formula (C) fails, when m + 1 = 0. Formula (D) fails, when p + 1=0. In these exceptionable cases the required integral can be obtained without the use of reduction formulae. 286 INTEGRAL CALCULUS 219. Derivation of Formula (A). Let us put for brevity X=a + baf, dX= nbx"'^ dx. Theu Cx'^X''dx= Cx'^-"+^-^'''^-^ Cx'"X''dx= Cx nb Integrating by parts with u = »;'""-"+', we have rx'"X^fe= ^""""'-^''"^ - "^7'^+^ Cx'-'^x^-^dx. . . (1) J nb{p + l) nb(p + l)J ^^ Comparing the integrals in (1), we see that not only is m diminished by n, but p is increased by 1. In order that p may remain unchanged, further transformation is necessary. By substituting X" ^-l = (a + bx") X", the last integral may be separated into two. Cx''-"X''+'^ dx = a Cx'^-^X" dx + b Cx'^X" dx. Substituting this in (1) and freeing from fractions, nb(p + 1) Cx'^X^ dx = a;»-''+iX''+i — (m — n + l)fa Cx'^-''X'' dx + b Cx X" dx Transposing the last integral to the first number,' (np + m + l)b Cx^X" dx = a;"'-»+iX''+i - (m — m + 1) a Cx'^-'^X" dx, (2) which immediately gives (A). INTEGRATION BY PARTS, REDUCTION FORMULAE 287 220. Derivation of Formula {B). Integrating by parts with u = X'', we have I x'^X^ dx = X"— I — pX'^'bnx"-^ dx ^^^^._j^p^r„,„^,_,^^_ . . . . (1) m + l m + lJ Comparing the integrals, we see that not only is p decreased by 1, but that m is increased by n. To avoid the change in m, substitute in the last integral of (1) hvS'=X—a. Also freeing from fractions, (m + 1) ^vTX^ dx = a!"+'X'' — np( Cx^X" dx — a Cx'^X"'^ dx\ Transposing to the first member the last integral but one, (np + m + 1) Cx'^X'' dx = x^+^X" + npa Cx'^X^^ dx, . . . (2) which immediately gives (B). 221. Derivation of Formula (0). This may be obtained from (2), Art. 219, by transposing the two integrals, and replacing throughout, m — n by m. This gives (m + 1) a Cx'^X" dx = af +^X''+' —(np + m + n + 1) Cx'^+"X'' dx, from which we obtain ((7). 222. Derivation of Formula (D). This may be obtained from (2), Art. 220, by transposing the two integrals, and replacing p—1 by p. This gives n(p + l)a ( x"'X^ dx=- k^+iX^+i + {np + n + m + l) Cx-^X^+^dx, from which we obtain (Z*). 288 INTEGRAL CALCULUS EXAMPLES 1. I — === — -Va — » +-^sin '-. Here C_^^_^ Cx'ia^-x'y^dx. Apply (^), making m = 2, ?i = 2, p = — -, a = a?, 6=— 1. Li = _?(a2_a!^)i + |sin-i^. 2. rVa' + a!'da; = |Va^ + a^ + flog(« + V?T?). Apply (5), making m = 0, n = 2, P=^, a = a', 6 = 1. Jia' + x^idx =^(a'+x')i + IJ- drr (a'+ xy = |(«' + ^')^ + 1 log(^ + ViH^- /^ dx -y/oc^ — a? , 1 _iX I = ; 1 sec -. J 3?^ a? — a^ 2 aV 2 a? a INTEGRATION BY PAETS. REDUCTION FORMULA 289 Apply (C), making m = — 3, n = 2, p= —-, a= — a', 6 = 1. = -^^ — sec — 2a?o^ 2a? a . C dx 1 1 _-,x 4. I = -sec^-. Apply (D), making 3 m = — 1, w = 2, j3 = — -, a= —a?, 6 = 1. Cx-\a?-aY^dx = - Aa^-q")"^ _ \Cx-\a? - a')-^dx 1 1 -iK — — -sec^- 5. rV^^^c?a; = ^V^^=^+^'sin-i^. »/ 2 2 a 7. r(a2-af)^da! = |(5a2-2a^VS^^=^ + 8. fCai^-tt^)^ dx = ^(2x'-5 aO V^^^+ ^* log (» + V^ 3a^sJQ-ix_ 8 a ■ma:'] 290 INTEGRAL CALCULUS •^ 8 8 a 10. fo^V¥+c? dx = ^(2a? + a") y/W+c^ - ^ log {x + V^Ta^). dx 11. r 12. Derive the formula of reduction used in Case IV of Rational Fractions. / dx _ 1 r X ,„ _ QN c d^ (x^ + a?Y ~2(n-l)a? (x" + a^)"-' '^^ '^ 'j (^2+ aV"' 1 o r dx 3af — cv' 3 _, a; 13. I - = ^^sec -. -^ 3?{x'-a^^ 2 aVVa^ - a^ 2 a= a 14- I , / , = Q^ Va^ + 1. ^^- J /o ^V2aa;-x^ + 2^vers->^. •^ V2 ax — ay' ^ 2 a Write I — — j , and apply (^) twice. ^ x-\/2 ax — 3? ■J V'2a — x Ifi r*- ^^ — V2aa; — a;^. x'\/2ax—v?dx C'^ 17. rV2aa;-x2dx = ^^V2ac(;-ar'+^%in-i- or = ^—^ V2 ax - CB^ + - vers-' ^' . ^ 2 a INTEGKATION BY PARTS. REDUCTION FORMULiE 291 Write J V2 ax-a?dx =J^'a^ - («= - <^f dx, and substitute in Ex. 6. 18. ra;V2^^3^fe=- ^"' + "''"-^'^ V2^^I^^ + ^%ers-'^ J 6 2 a ,Q rV2 ax — x'dx f^ 5 , _ia;. 19. I = V 2 aa; — cs^ + a vers^ - J X a 20 r "'"'^^ _ _ a!'°~ V2 go; - !«^ (2 m — l)a r a!°-^c;g; ^ V2 a.s — a^ «i m J V2 aa; - c da; 21. f-- a;"" V2 ax— XT' '9 «^._^ ■\/2 aa; — x^ . m — 1 /» da? (2 m- l)aa;» (2^iirn)^J ar-V2aa;-^' 22. ra;'"V2aar^da; a;'-'(2 aa; - x^)' , (2 m + l)a f ^^ ^^^ -„ ^ = ^^ j: — ^ + '^ —^ I a!"^' V2 ax — ix^ dx. m + 2 m + 2 ^ 23. JV2] I ax — a;^ da; ^ (2ag — af)^ w-3 r V2ax-g:^da; . " (2 m — 3)aa;'» (2 w - 3)aJ x"-! 223. Trigonometric Reduction Formulae. — The methods explained in Arts. 211, 214 are applicable only in certain cases. By means of the following formulae, J sin" x cos" X dx, j tan"* x sec" x dx, and | cot" x cosec" x dx, may be obtained for all integral values of m and n, by successive reduction. sin"'+' X cos" +' ' w + 1 ' n + ; 292 INTEGRAL CALCULUS fsin" X cos" xdx = - ^^'^""^ '^ ^°^""^' '^ + ^^^i^Ill fsin^-^aJCOS^Kda;. (1) rsiii"a;cos"a;da; = ^™'""^'^"°^""''" + ^Ji^ fsin'" a; cos"-' a; da;. . (2) ♦^ m + w m + «»/ j sin" a; cos" a; da; sin"+ia;eos"+'a; , m + n + 2 r . „.„ „ , .on = — — ^— i sin'"+2a;cos"a;da;. . . . (3) m+1 m+1 J ^ ' I sin™ a; cos" a; cZa; 3os"+^a; , m + m + 2 r . „ „4.o , /.. — 1 — ^ I sin"a;cos''+'a;da;. . . (4) ■ 1 71 + 1 «/ /^■■.„m„,A^ sin""' X COS a; ,m — \C- m-z ^ ,f^ sin" a; da; = 1 I wa^ 'xdx (6) m m J fcos" X dx = '^ ^ '^°^''"' '^ + !il^ fcos"- a; da; (6) ftan- a; sec" a; da; = tan"-' a; sec" a; m-1 A g^j^„_2 ^ ^^^„ ^ ^^ .j. J m + m — 1 m + w — 1»/ ^ ' / __ cc m+n—1 m+n J„ J sec"-' a; tan a; , ?i — 2 /^ „ „ , see" a; da; = 1 -■ I sec"-' a; da; (Q) n—1 n—lJ ^ ' cot" x cosec" x dx cot^-'a; cosec" x m — 1 T , „ „ „ , ,„. - I cot"*-' a; cosec" x dx. . (8) INTEGRATION BY PARTS. REDUCTION PORMULiE 293 fcosec"a;da; = -52!£2!l^^2t^ + ??iZL2 rcosee»-2a;da;. . . .(10) *J n — ln — lJ Ctan''xdx = ^^^''~''^ - Ctan''-^xdx (11) Ccot"xdx = -?^^^- Ccof'-^xdx (12) 224. Derivation of the Preceding Formulae. — To derive (1), we integrate by parts with u = sin""' x. fsin"* X cos" xdx = - sm'°-^g'Cos"+^a; _^ m-1 T gjj^™-. ^ gog„+2 y. ^^_ J n+1 n+lJ I sin""^ X cos""''^ X dx= I sin"""^ x cos" xdx— | sin" a; cos" a; dx. Substituting this in the preceding equation, and freeing from fractions, we have (m + »i) I sin" X cos" x dx = — sin"~' a; cos"+' a; + (m — 1) | sin"*"^ a; cos" a; da;, which gives (1). To derive (2), integrate by parts with u = cos"~^ x, and proceed as in the derivation of (1). Formula (3) may be derived from (1) by transposing the integrals, and replacing m — 2 by m. Formula (4) may be derived from (2) by transposing the integrals, and replacing m — 2 by n. To derive (5), mg,ke m = ,0 in (1); and to derive (6), make m = in (2). The derivation of (7), (8), (9), and (10) is left to the student. We have already derived (11) and (12) in Art. 212. 294 INTEGRAL CALCULUS EXAMPLES 1 r„;„6„j™ cosw/sin^o; , 5 . , , 5 ■ \ , 5 J 4 l^sin^a; 2siii2a;y 8 ^ 3. rsec^xdx=^^f^—+ ^ +g ^ 2 cos^ a; V 3 cos'' a; 12 cos^ a; 8 5 + — log (sec a; 4- tan a;) . 4. fcoss a; dK = ?i5^ f'cos' x + - cos' a; + — cos= a; + — cos 9;^ + — J 8 \ 6 24 16 y 128 E /^ • 4 2 J cos a; /sin* a; sin^a; sinaiX , x 5. I sin'a;cos^a;da;= ( )H — J 2>y^3 12 8;ie e /^cos*a;, Scosx — 4 cos'a; 3 cos a; , 3, , 6. I dx= ^- — 1- -log tan J sur X 4 sin* x 8 sm^ x 8 ^ r dx ^ 1/1 J sin* a; cos' a; cos^a;\3sii 16 , 6 5 . \ sin*a;cos'a; cos^a;V3sin'a; 3sina; 2 J 5 + - log (sec X + tan a;). Li 8. ftan^a; sec' a; dx = f^-^ - ^-^\ sec' x + ? /tan' X tan a;\ , , sec x tan a; 16 + — log (sec X ,+ tan x). fcot' X cosec" aida; = ''"'^ ^ cosec a; / cosec* x cosec^a; 1 J 2 V 3 12 ^8 — -—log tan-- 16 ^ 2 CHAPTER XXVI INTEGRATION BY SUBSTITUTION 225. The substitution of a new variable has been used in Chapter XXIII, for the rationalization of certain irrational integrals. We shall consider iu this chapter some other cases where, by a change of variable, a given integral may be made to depend upon a new variable of simpler form. We shall first consider some substitutions applicable to integrals of algebraic functions, and afterward those applicable to integrals of trigonometric functions. f{3?)x dx, containing (a + &af)«. One of the most obvious substitutions, when applicable, is ay' = z. By this, any integral of the form if(oif)xdx is changed into - )/(«)< )dz. p Integrals containing (a + bx^i are often of this form. a^dx Take for example | - By the substitution se' = z, idz / a?dx ^1 r zd This is of the form of Art. 203, and is rationalized by putting l—z=w\ 295 296 INTEGRAL CALCULUS The two substitutions in succession are equivalent to the single substitution 1 — x^ — vf. Applying this to the given integral, a^ = 1 — w^, xdx=^ — 'w dw. = -(— f)=-|(3-»^ = -^^\a^ + 2) EXAMPLES 1_ r a^dx ^ 3x*-2x-^ + 2 ^^^^:^ J V2a;2+1 30 2. Ca^ia? - oif)idx = A.(6 x' - aV -Ba^Xa' - af)*. 3. 1 ^^ ^J^iogV «'' + «' -_^ = ^1og «=" •>' a; Va;^ 4- a^ 2 a Vaj^ + a^ 2a Va;^ + a' + a 2a • (Va;^ + a= + a)^ 1 T X = - log - « V?+T' + a 227. Integration of Expressions containing -\/a? — a? or Va;^ ± a^, by a Trigonometric Substitution. Frequently the shortest method of treating such integrals is to change the variable as follows : INTEGRATION BY SUBSTITUTION 297 For Va^ — ^, let a! = asm0 or a! = acos^. For Va^ + o?, let a; = atan0 or a! = acot^. For Var^ — a^, let x=a sec ^ or a; = a cosec 6. For example, find ( — Let a; = a sin ^, dx = a cos 6 d0, a^ — a:^ = a^ — a^ siij2 ^ _ (j2 cqs^ ^. dfl tan X r da; ^ r acosedg ^ 1 T (ap — a^i J a^cos^e a^J cos^^ a' ap^a' — x' Take for another example i - x-yjoi? + a^ Let x=^a tan 6. / " da; ^ r asee^ede _1 Tsece^^^ 1 r dg J a; Va;^ + a^ *^ atan^-asec^ aJ tanfl aJ sin^ = - log (cosec ^ — cot 6) = - log • a a a; Again, find | ■ Let a; = a sec 6. '^^^dx. r^yZE^dx^ ptang-asecgtangde^^ ftan^fld^ J a; ^ a sec 9 J = a r(sec= e-V)d6 = a (tan ^ - e) = V«^^-a^ — a see"' - • a 298 INTEGRAL CALCULUS EXAMPLES 1 j Va^ — a^ da; = - Va^ — a;^ -{- a' . _i a; 2 a dx ^a? + c? dx Vx^ — a" = log (a; + -y/af — a^). r dx ^ _ (2a;^ + «^V«--g' , >/ Va;^ + a' dx = - Vx' + f ■ + log(a; + Va;^ + a^- 6. ^v?E«!d,=(^!^i^^ J a;* 3a^ar= y r da: ^ (2a;'''-l)V^M^ . 8- r\/^-^'^a'= f- ^-^^^ dx = V^^^^ + log (x + VB?3T). ^ \a;-l -' Va:^-1 X--1 (a! + l)VS^^^ ^a; + l 10. /, dx (l-a;)Vl-x2 = ^l'-^^ ^1-x "■/ a; da; = -V3 + 2a;-a;2^gijj-i«j3l. V3 + 2 a; - a;2 2 INTEGRATION BY SUBSTITUTION 299 For V3 + 2a;-a^= V4-(a;-l)^, let a;-! =2sin^. 12 ■/; dx x + 1 (x' + 2x+3y 2Va^ + 2a; + 3 For Va^+2 a; + 3 = V(a;+ 1)^ + 2, let aj + 1 = V2 tan «. 13 C——^ — a; — a "^ {2ax — x^i aV2 aa; — a^ 14. fJlE^dx = r-4^^ da, = VS^=^ + ^ sin- ^^^^ • 15. r* a;^ c'a; _ (3 a + x) V2 aa; — a^ 3a^ ■ _i a; — a "^ V2 aa; - a;^ ~ 2 2 a 228. Substitutions for the Integration of Trigonometric Functions. A trigonometric function can often be integrated by transforming it, by a change of variable, into an algebraic function. For this purpose two methods of substitution may be used, as shown in the two follow- ing articles. 229. Substitution, sin x=z, cos x = z, or tan x = z. Consider, for example, /* sin x cos x dx J 1 — sin a; + cos^ x dz Let sin x = z, then x = sin~' z, dx = VI -z^ / sin x cos xdx _ C zVl — z^ dz _ r zdz 1 — sin X + cos^ X J l—z + l—z^ Vl— ^ "^ 2 — z — z^ / zdz _ 2 r dz 1 r dz (2 + 2)(l-2) 3J2 + 2 3Jl-« = -|log (2 + 2)-| log (l-2)=-| log[(2 + sina;y(l-sina;)]. 300 INTEGRAL CALCULUS J a' dx (2 1-nn2 /v. /i2 7j2 +b^tan^x a' EXAMPLES X — tan"-' I - tan x a \a ^•/r^.=i+l^°^(^^'^^+^°^^)- 3. r^^^dx = llogl^^^+J-\og^ J sm 4tx 8 1 + sin x 4. ■y/2 1 1 1 1 + V2sins ■ V2 sin a; Let sin x = z. f dx ^ll^gsmxil + cosx) j^g^ cosx^z. sin a; + sin 2 a; 3 (1 + 2 cos a;) 5 rsinx + cosx d^^3£_liog(sin^ + 2cosa;). J sin X.+ 2 cos a; 5 5 Let tan x = z. 6. P^^^dx = x + ^log J tan a; VS tan X — VS V3 "tana;+V3' 7. Show, by transforming into algebraic functions, that only one of the following integrals can be expressed in terms of the elementary functions. (See Art. 208.) I Vtana; dx= \ — ^zdz where z = tan x. //—■ — _, C 'Vzdz r zdz , Vsina;aa;= I — = I — , where z = si sin a;. INTEGRATION BY SUBSTITUTION 301 230. The Rational Substitution, tan 5 = 2. By this substitution, sin X, cos X, tan x, and dx are expressed rationally in terms of z. For 2 tan? sin T — "^ 22 1 + tan^l 1+2'' 1-tan^l (>0R T — 1-2^ 1+tan^^ 2 "1+2^ 2tan- , 2 tan X = = 1-tan^l 2z 1-2^ - = tan"^2. da ._ 2d? From It follows that the integral of any trigonometric function of x, not containing radicals, may be made to depend upon the integral of a rational function of 2, and can therefore be expressed in terms of elementary functions of x. : Applying the substitution of the a + 6 sm a; ^ *^ preceding article, tan ^ = z, 2dz 2dz /I z az _1±Z_= C- . 2bz J a (1+2^+262 1+2^ / 2adz _ r 2 adz a'z' + 2abz + a'~J (az + by + a'-l atan -+ b 302 INTEGRAL CALCULUS If a > 6, numerically, r ^^ = ^ tan- «' ^ + ^ = 2 tan-'lIlL_ If a < &, numerically, da; r 2 adz 1 , „a» + & — V&^ — a^ / da; _ r a + b sin a; J (( log- + 6sina; J («» + 6)2 _ (6^ _ «=) V&^ZI^s ag + ft + Vft'-a' a tan - + 6 — Vb^ — a' 1 , a log ^*'-'*' atan'^+6+V6^^^^ '2 232. To find C- '^^ J a + b cos X 2dz / • da; ^ / 1+g' r Ja + 6cosa; f , b(l—z'^~J (a 2dz b)z'-\-a + b =— ?- r tZz a — bJ o , a + 6 a — b If a > 6, numerically. J o + 6cosa; a— &»a+6 Va + fe = — — =r tan~Y\/ tan - INTEGRATION BY SUBSTITUTION 308 If a < 6, numerically, dx 2 r d^ / dx _ _ 2 r a + b cos X b — a J . b — a 1 J gV6-a-V6 + a V&^ — a? z V6 — a + V6 + a J Vft — a tan 5 + V6 + a , log Vo — a tan k — Vo + a EXAMPLES Integrate the following functions by means of the rational sub- stitution. J,_^ = ltan-ir2tan'^" 6-3cosa; 2 V 2; dx 1 T tana; + 2— V3 I. C ^ = ^-log J l + 2sin2a5 2V3 + 2sin2a5 2V3 tana; + 2+A/3 3tan^ + 2 J. f ^^ = lw ? J 5 sin a; + 12 cos x 13 ° 2 tan - — 3 Li Stan a+ Va^ + h^ , Jasina; + 6cosa; -^W^' % tan ^ - a - V^H^^ tan- c?a; i„„ 2 f-^-^ = log J sm a; + vers x sm 05 + vers a; i + tan^ 2 304 INTEGRAL CALCULUS '■/: '^^ ■ = taii-i. tan — 1 3 — sin X + 2 cos x 3tan- + l dx 1 , 2 r. r ^i5 =llog J 5 + 7 sin a; — cos x 5 5 + 7 sin a; — cos x 5 ^ x , r, tan - + 2 2 /- = - tan — — log ( 1 + tan - ] • (l + sma! + cosK/ 2 2 -^j^^^^^ \ 2j 233. Miscellaneous Substitutions. Various substitutions applicable to certain cases will be suggested by experience. The reciprocal substitution, x = -, may be mentioned as simplify- ing many integrals. EXAMPLES Apply the reciprocal substitution a; = - to Exs. 1-6. 2^ ^- J X* 3a'x' of ^^ Var^ + a' x^y/oi? + a^ 3 r -^2ax-x^ ^^^ i2ax-a^^ _ ' J of Sase' i. f— ^^= = -log ^ •^ a;Va^±a^ « a + Va'±a;2 ■ J a;* 8 a;* 6. X INTEGRATION BY SUBSTITUTION dx 305 • _i a; — 1 ^ = sm ' a'V8a;2 + 2a;-l 3 as / a^dx 8 {x + 2y 3(a; + 2)3 (a; + 2)2 a; + 2 ^ +-^ + log(a; + 2). Let x + 2=z. J ^ ^^ ^ w + 2 ^ ^ n + 1 Let w + 6 = 2. 9 r dx 1 J x(a + bxY a' bV 2bx ^^ a + bx' _2(a + bxy a + bx x Let a+bx = xz. Jti 10. I — 5iil^_da; = sina;-sinalogtan^4^- Jjet x+a = z. tan (x + a) 2 "Js <^^ .a;-e-"^ + logVe^ + e'= + l — tan- '^^'' + ^ - = + e^ + e* V3 V3 Let e" = z. 12. r ja-x^^ = V(a - a;) (6 + a;) + (a + &) sin"' J^ + & + b Substitute b + x = z^, and tlie integral takes the form of Ex. ,5, Art. 222. 13. fJ^dx •^ ^x + b = V(a; + a)(x + &) + (a— 6)log (Vx + a + Vx +b). Substitute x + b = z^, and the integral takes the form of Ex. 2, Art. 222. CHAPTER XXVII INTEGRATION AS A SUMMATION. DEFINITE INTEGRAL 234. Integral the Limit of a Sum. An integral may be regarded and defined as the limit of a sum of a series of terms, and it is in this form that integration is most readily applied to practical problems. 235. Area of curve the limit of a sum of rectangles. Let it be re- quired to find the area PABQ included between the given curve OS, the axis of X, and the ordi- nates AP and BQ. Let y = x^ be the equa- tion of the given curve. Let OA=a, and OB=h. Suppose AB divided into n equal parts (in the figure, n = 6), and let A« denote one of the equal parts, AA^, Then AB =b — a = n ^x. At Ai, A.„ ■■■, draw the ordinates A-^Py, A2P2, •••, and complete the rectangles PA^, P1A2, From the equation of the curve y = x^, PA = J, Pi^i = (a -f Aa;)*, P^A^ ={a + 2Ax)i, Area of rectangle PA^ = PA x AA^ = a^Aa;. Area of rectangle PiA^ = PiAi x A1A2 = (a + Ax)^ Ak. Area of rectangle P2^3 = -P'2^2 X A^A^ = (a + 2 Aa;)^ Aa;. Ai Ao •, QB=b^ Area of rectangle P^B ■■ ■.P,A,xA,B-. 306 : (6 — Aa;)^ Aa;. INTEGRATION AS A SUMMATION. 307 The sum of all the n rectangles is ah Ao; + (a + Aa;)i Aa; + (a + 2Aa;)^ AxH h (6 - Aa;)* Aa?, which may be represented by V x^ Aa;, where x^ Ax represents each term of the series, x taking in succes- sion the values a, a + Aa;, a + 2 Ax, •••, b — Aa;. It is evident that the area PABQ is the limit of the sum of the rectangles, as n increases, 'and Aa; decreases. That is, Area PABQ = Lim^^^„ V x^ Aa;. ' a 236. Definite Integral. From the preceding article V a;^ Aa; = a^ Ax +(a + Aa;)i Aa; + (a + 2 Ax)^Ax-| 1-(& — Ax)^ Ax. The limits of this sum, as Ax approaches zerp, is denoted by I X* dx. That is, by definition, i * 1 X~N^ 1 x^dx:= Lim^^o > x^ Aa;. X X* dx is called the definite integral, from a to b, of x^ dx. It is to be noticed that a new definition is thus given to the sym- bol I , which has been previously defined as an anti-differential. The relation between these two definitions wUl be shown in the fol- lowing article. 237. Evaluation of the Definite Integral I x^ dx. This is effected by finding a function whose derivative is x* x^ = ^f'-f)- dx\ 3 By the definition of derivative, Art. 15, 2, , . .4 2xi , -(x+Ax)^ dx\ 3 J '^ Ax 308 INTEGRAL CALCULUS Hence 3^'' + ^'') "T 2/™ , a„nI 2a;^ = a;* + £, Ax where e is a quantity that vanishes with A*. 9 a 2 a;^' J. Hence -(x+Ao;)^ — =»- Ax + tAa;. o o Substituting in this equation successively for x, a, a + Ax, a + 2 Ax, ■■■,b — Ax, 3 ? (a + Ak)4 - ^ = a* Ax + £i Aa;, = (a + 2 a;)4 — - (a + Ax)i = (a + A.r)^ Aa; + e^ Ax, o o -(a + 3Ax)^-^(a + 2Ax)i={a + 2Ax)iAx + csAx, o o 2^ _ ? (6 _ Ax)' = (6 - Aa;)*Ax + e„ Ax. o o Adding and cancelling terms in the first members, 2&!_2_^=a^Ax + (a + Ax)^Ax + (a + 2A9;)*AxH l-(6-Ax)^Ax o o + ej Aa; + £2 Ax + €j Ax + • • • + e„ Ax = Tx^Ax + TeAx (1) Comparing with the figure, Art. 235, V x- Ax, as we have before seen, represents the sum of the rectangles, and X e Aa; represents a the sum of the triangular-shaped areas between these rectangles and the curve. The latter sum approaches the limit zero, as Ax approaches zero. INTEGRATION AS A SUMMATION 309 For if £i IS the greatest of the quantities t^, e^, ••■£„, it follows that ' a ^^ a that is, y £ Ax < £t(6 - a). As £j vanishes with Ax, LiuQa^X £Aa; = 0. a Taking the limit of (1), 2 6^ 2a^ T- V'iA that is, Cx^ dx=~-— = Area PABQ. X' 1 a;^ dx is found from the integral pax='^, by substituting for x, b and a in succession, thus giving 2bi _2ai 3 3 The process may be expressed •/» 3 „ 3 3 This is called integrating between limits, the initial value a of the variable being the lower limit, and the final value b the upper limit. In contradistinction is called the indefinite integral of x^dx. 310 INTEGRAL CALCULUS 238. General Definition of Definite Integral. In general if f{x) is a given function of x which is continuous from a to h, inclusive, the definite integral r /(a;)da; = LiniAa^o /(a)Aa; +/(a + Aa;)Aa; +/(a + 2 Aa;)Aa! + - +/(6 - Aa;)Aa! 1 If \ f{x)dx = F (x), the indefinite integral, rfix)dx = F(b)-Fia) (1) This may be illustrated by the area bounded by a curve as in Art. 235, by supposing y=f(x) to be the equation of the curve OS. The proof of Art. 237 may be similarly generalized by substitut- 1 2 x^ mgf(x) for a;*, and F(x) for — ^ ■ Geometrically the definite integral I f(x)dx denotes the area swept over by the ordinate of a point of the curve ?/=/(»), as x varies from 6 to a. It is to be noticed that in Art. 192, by a somewhat different course of reasoning, we have arrived at the same result. Area PABQ = ^(6) - F(a). 239. Constant of Integration. It is to be noticed that the arbitrary constant G in the indefinite integral disappears in the definite integral. Thus, if in evaluating I x^ dx, we take for the indefinite integral s t , 2xi , ^ x^ dx = — — h C, o wefind j^.Ux = -+C-{-^ + Oy—--. Or if ^f{x) dx=F (x) + C, Cf{x)dx = F{h) + G- \F{a) + Cr\ = F{b)--F{a). .INTEGRATION AS A SUMMATION 811 EXAMPLES 1. Compute X a^ Aa; for different values of Ax. When Aa; = .2, ^x'Ax = (1^ + o' + n' + !:& + r8')(.2) = 2.04. When Aa; = .1, ^x'^Ax = (12 + n' + 1:2' + . . + i:9')(.l)=2.18. When Aa; = .05, ^^ ^Aa; = 2.26. Lim^^oV a;^ Aa; = | 3^ dx = 8-1 = 2.33. Curve OS, y=a?. 0A=1, 0B=2. Area PABQ = 2.33 square units. x~v*Aa; 2. Compute X — for different values of Aa;. ■^^1 X When Aa; = 1, When Aa; = .5, > — = ■*^i X 1.593. WhenAa;=.l, V — = 1.426. ^-11 a? 812 INTEGRAL CALCULUS Lim^^ V'— = f'^ = log a; ' = Irfg 4 - log 1 = log 4 = 1.386. ^i X Jy X 1 Curve EQ,y=:-- OA = 1, 0B = 4. Area PABQ = 1.386 square units. 3. Compute X ^ ^^> when Aa; = 1 ; when Aa; = .6 ; when Ax = .2. ' Ans. 18; 19; 19.6. rind Lim^3,^o z] *^ '^^^ ^*''*- ^'^■ 4. Compute V ^, when Aa;=.2; when Aa;=.l; when Aa;=.06. » 1 + « ^^^ .833; .810; .798. FindLim^^y'-^. Ans. ^=.785. -^0 1 + ar 4 5. Computed logmajAa;, when Aa;=l; when Ax = .5; when Aa;=.3. '" JlriS. 3.121; 3.150; 3.161. Find LimAi=o Zj logjoajAa;. ^ns. 13 logiol3— 3 logioe— 10=3.177. 6. Compute y tan^A<^, when A<^=3°=— ; whenAi^ = — ; when Ans. .316; .328; .340. Ans. log. V2 = .346. p = 180 TT rind Lim^,^ V tan 4 <^A<^. 7. C\x'-iyxdx = 125 6 9. f dy _ TT 11. f^^'^-- tan- Jo e^ + 1 4 '"' a; da; 8. r — '• " ""^ = 2. 10. J' -^^ ar^ - a; + 1 3 ^3 8a'(7a5 12. r ;^^^, = 2,ral »/o ar + 4 a^ INTEGRATION AS A SUMMATION 313 13. /»3 f^-g w J^ Vta;-2)(3-a;) 14- £ siu' tf, d = —■ 15. TT £sin^6de = ^-l. ^g cKe.s^20de = l- 12 Jo 4: 17. r'*^°^2e-cos2«^^_j^^^^^^_ Jo COS 6 — COS a 3 18. r ^'^'^ -17 6 log 2. Jo (a; + 2)' 4 ^ 19. C^—. .''!..„. .=Tlog(2e). rV3 16 20. I x^ sin a; da; = jr — 2. 21. r'"xlog(a; + a)dx = ^log(3a)-^. 22. f tan-i -dx = tt — log 4. Jo 4 ,^)(ar' + 62) ^a + 6V4 Ja (x'-{- a"" ' " ' '' ^ By (5) aud (6), Art. 223, we find TT TT J -2 ?i — 1 r^ sin"a;dx = I sin''-'''xdx, n Jo TT ^ cos" a; da; = I cos""^a;da;; n Jo from which derive the following results : 314 INTEGRAL CALCULUS 24. If n is even, — E Jo Jo 2-4-6"-n 2 25. If n is odd, Jsin''xdx= I ^0 sin" xdx= [ cos" a; da; = "'^ ^• 3-0-7 •■•n 240. Sign of Definite Integral. In considering the definite inte- gral I / (a;) dx, we have supposed a < 6, and / (x) to be positive be- tween the limits a and b. If /(a;) is negative from x = a to x = b, ^ f(x) Aa;, being the sum of a series of negative terms, is negative, and consequently Jf (x) dx is negative. If f{x) changes sign between x = a and x = b, I f{x) dx is the algebraic sum of a positive and a negative quantity. Y For example, | cos x dx = 1 = area AOB. I cos a; dx = — 2 = area BCD. 2 INTEGRATION AS A SUMMATION 315 Sir I COS xdx=—l = l — 2. Jf2n COS a; da; = = 1 — 2 + 1. The change of sign resulting from a < 6 is considered in Art. 243. 241. Infinite Limits. In the definition of a definite integral the limits have been assumed to be finite. When one of the limits is infinite, the expression may be thus defined : /(a;) da; = Limj^ I f(x)dx. For example, consider Ex. 12, following Art. 239. Jo of + 4: or Jo or + 4: or \ 2a J Eeferring to Art. 126, we find the geometrical interpretation of this result. The area included between the curve, the axes of X and Y, and a variable ordinate, approaches the limit 2Tra^, as the distance of the ordinate is indefinitely increased. X'°dx — , we find J'*' dx — = Limj^ log 6 = oo . I X In this case there is no limit, and the expression | — has no My I X meaning. 242. Infinite Values of /(«). In the definition of J f{x)dx,f{x) is assumed to be a continuous function from a; = a to a; = &. If f{x) is continuous for all values from a to b except x=a, where it is infinite, the definite integral may be defined thus : J/(a;) da; = Lim^^o ( f{x)dx. 316 INTEGRAL CALCULUS If /(6) = 00, /(x) being continuous for other values of x, f{x)dx = JAm^^ I f(x)dx. For example, consider Ex. 9, following Art. 239, r di/ _ Here — =^:::^ = 00 , when y = a. Va^ — y^ Hence I y =Limt^o ( ^ =Lima^(sin-'^^ ^) "■2~6"3' Another example is Ex. 13, following Art. 239. dx 2 V(a;-2)(3-a;) Here — =^^=:^^^ = oo, when a; = 2, and also when a; = 3. V(a;-2)(3-a;) „ C^ dx -r- r""" f?'« Hence I — z^^^=^^^=. — Limj=o I — ^^= -'^ V(a;-2)(3-a;) *^2+« V(a;-2)(c : Lim^^o sin"' (2 a; — 6) V(a;-2)(3-,T) '= Lim^^ [sin-' (1-2 h) - sin-' (- 1 + 2 A)] A = sin-'l — sin~^(— l) = 'jr. If f{x) is infinite for some value c between a and b, and is con- tinuous for other values, the definite integral should be separated into two. r f{x) dx = f f{x) dx + ffix) dx. See Art. 243. These new definite integrals may be treated as already explained. INTEGRATION AS A SUMMATION 317 EXAMPLES r dx ^TT 2 r (l+af)^ 4 ■ J„ (a!^ + a=')(a;2 + 62) 2a6(a + 6) f ^'^ =:log(2+V3). 4. r_^^ = llog2. 243. Change of Limits. The sign of a definite integral is changed by the transposition of the limits, I /(a;)da; = — | f(x)dx. This is evident from (1), Art. 238, and also from the definition. For if x varies from b to a, the sign of Ax is opposite to that where x varies from a to 6. Hence the signs of all the terms of 2 /(*) ^^ will be changed, if the limits a and b are transposed. J '•a /»6 f{x)dx = — I fix) dx. A definite integral may be separated into two or more definite integrals by the relation, I f(x)dx= I f{x)dx+ I f{x)dx. This follows directly from the definition. 244. Change of Limits for a Change of Variable. When a new variable is used in obtaining the indefinite integral, we may avoid returning to the original variable, by changing the limits to corre- spond with the new variable. For example, to evaluate n dx •^0 1 + Va; assume y/x = i 318 Then we have INTEGRAL CALCULUS dx 2zdz 1+y/x 1 + 2 Now when x = i, z = 2; and when a; = 0, 2 = 0. Hence C-^^- ^ C ^1^ = 2 [z -log {1+ «)] : 4 - 2 log 3. EXAMPLES 1. rxy/^rr2dx=^. Ji 15 a. C(x-2yxdx = - ^'^ + ^ Let X+2-. -^ r/ m^ + 3 n + 2 Let a; — 2 = 2. 3. r "^^'^ =% (3 + ^)- Let a^ + l=2^ h (a;2 + 1)* 8 J-»2o V2 ax — 3?dx = a ■KCt 4 Let a; — a = a sin B. 5. r V3 + 2 (g - a:' (^a; = V3 - ;. Let a; = l+2sine. ji (x — ly 3 6- J^ V(a;-a)(6— a;)da; = ^ (6 — a)^ Let a; = a cos^ <^ + 6 sin^ <^. 7. r(a*-a;J)^da; = 5^. Let a; = a sin' 6. 8. r9:4V„2_^(ja.^/:^ + ^\„6_ -Le^ a; = asin6l. •^°. \ 64 48/ INTEGRATION AS A SUMMATION 319 245. Definite Integral as a Sum. In the application of integration it is often convenient, in forming the definite integral from the data of the problem, to regard | /(as) dx as the sum of an infinite number of infinitely small terms, f(x)dx being called an element of the required definite integral. From this point of view, ff(x) dx =/(a) dx + f(a + dx) dx + f(a + 2 dx) dx+ ••• +f{b) dx. This may be regarded as an abbreviation of the definition of a definite integral given in Art. 238. CHAPTER XXVIII APPLICATION OF INTEGRATION TO PLANE CURVES. APPLICATION TO CERTAIN VOLUMES 246. Areas of Curves. Rectangular Coordinates. We have already used this problem as an illustration of a definite integral. We will now consider it more generally, and derive the formula for the area in rectangular coordinates. 247. To find the area between a given curve, the axis of X, and two given ordinates AF and BQ ; that is, to find the area generated by the ordinate moving from AP to BQ. Let OA = a, OB=h. Let X and y be the coordi- nates of any point P^ of the curve; then x + ^x, 2/ + A2/, will be the coordinates of P3. The area of the rectangle P^A^As is PiAi X -42^3 = y Aa;.* A4 B The sum of all the rectangles PAAi, P1A1A2, P^A^A^, • • •, may be represented by ^ ^2/ ^■'^• The required area PQBA is the limit of the sum of the rectangles, as Ax is indefinitely diminished. That is - = J ydx, * By Art. 245, one readily sees that this rectangle is an element of area. 320 APPLICATION OF INTEGRATION TO PLANE CURVES 321 the lower limit a = OA, being the initial value of a;, and the upper limit b = OB, the final value of x. Similarly the area between the curve, the axis of Y, and two given abscissas, GP and HQ, is A=^ \ xdy, the limits of integration being the initial and final values of y, g=OG, a.ndh=OH. EXAMPLES 1. Find the area between the parabola y^ = 4:ax and the axis of X, from the origin to the ordinate at the point (h, k). Here A=C ydx= C2a^^^ *^0 Jo dx 4 a^x^ I * 4 ahi^ Since k^=i.ah, k=2 aVi^, which gives A = -h2aih^ = -hk = -0MP]Sr* 3 3 3 2. Find the area of the ellipse ^ + t = l. a" b' dx Area BOA = I ydx = - I Va^— x' «/o Ct»/o a\_2 2 a _ irab ~ 4 * In finding areas, after the element of area and the limits of integration are chosen, the problem becomes purely mechanical. 322 INTEGRAL CALCULUS The entire area = irob. Or we may integrate by letting x^a sin c^. Then C^a^- a?dx = aj' C sin^ d=^ p(l -cos2(^)d<^=^. Area 50^ = ^.!^' = ^. a 4 4 3. Find the area included between the parabola a;^ = 4 ay, and the witch y = 8a= a;^ + 4 a^ Ans. 2w--W. Having found the point of intersection P, (2 a, a), we proceed as follows : Area AOP= AOMP- OMP* Jo 9;2 + 4a2 Jo 4o 3 Area between two curves = ( 2 tt — - la', 4. rind the area of the parabola {y-5f = S{2-x), on the right of the axis of Y. * Length of element of area is the y of the witch minus the y of the parabola. Ans. 10|. APPLICATION OF INTEGRATION TO PLANE CURVES 323 5. Show that the area of a sector of the equilateral hyperbola 3? — y^ = a?, included between the axis of X and a diameter through ft O* I 7/ the point {x, y) of the curve, is ■— log """^ • fcc\ ^ f'u\ " 6. Find the entire area within the curve (Art. 133) (-)+(- 1 =1. \aj \bj Ans. - irab. 4 7. Find the entire area within the hypocyoloid (Art. 132) »^+2/'=*- Let aj = a sin' <^. Ans. ^ 8. Find the entire area between the cissoid (Art. 125) y^ = ; 8 2a — X and the line a; = 2 o, its asymptote. Ans. 3 vo'. 9. Find the area of one loop of the curve (Art. 133) a*y^ = d?x^ — «*. Also from x = - to a; = a. Ans. l--\ — —- ] — . 2 \3 8 y 4 10. Find the area of the evolute of the ellipse (Art. 167) (ax)^ + (by)i=(a'-b^l Ans. ^(|-^ 11. What is the ratio between a and b, when the areas of the ellipse and its evolute are equal ? Ans. g^ Vg+V2 ^2.11. 6 V3 12. Find the area included between the parabolas y^=:ax aad x^ — by. Ans. — - 324 INTEGRAL. CALCULUS 13. Find the area included between the parabola 2/^ = 2 a; and the circle 2/^ = 4 a; — a^. Ans. 0.475. 14. Find the area included between the parabola 2/2 = 4 aa; and its evolute (Art. 167) 27 ay^ = 4 (a; - 2 af. _ Ans. ^^a^ 15 Parametric Equations. Instead of a single equation between x and y for the equation of a curve, the relation between x and y may be ex- pressed by means of a third variable. Thus the equations a; = a sin ^, y = a cos ^, (1) represent a circle ; for if we eliminate (j> from (1) we have a!^ + 2/^ = or (sin^ 1^ + cos^ ) = a'. Equations (1) are called the parametric equations of the circle, and the third variable is called the parameter. The formula A= I ydx is applied to (1) by substituting 2/ = a cos , dx = a cos <^ d. For a quadrant of the circle A = ] y dx = I a^ cos^ ^d4> = ^ . 15. Find the area of one arch of the cycloid a; = a (^ — sin 6), y =a(l — cosO). Ans. Zird^. 16. The parametric equations of the trochoid, described by a point at distance 6 from the centre of a circle, radius a, which rolls upon a straight line, are x = ad — hs\n6, y = a — bcosd. Find the area of one arch of the trochoid above the tangent at the lowest points of the curve. Ans. Tr(2 a+b)b, when 6 < a or 6 > a. APPLICATION OF INTEGRATION TO PLANE CURVES 325 248. Areas of Curves. Polar Coordinates. To find the Area POQ included between a Given Curve PQ and Two Given Radii Vectores ; that is, to find the area generated by the radius vector turning from OP to oq. Let POX=a, Q0X = I3. Let r and be the coordinates of any point Pj of the curve, then r + Ar, e + M, " will be the coordinates of P^. The area of the circular sector P^OB^ is The sum of the sectors POB, PiORi, P2OB2, •••, may be rep- resented by .2' V^lr'AO. The required area POQ is the limit of the sum of the sectors, as A6 approaches zero. That is. = 11^''' the initial value of $, a = POX, being the lower limit, and the final value of ^, /8= QOX, the upper limit. EXAMPLES 1. Find the area of one loop of the curve (Art. 144) r = a sin 2 6. ^ = i f V de = ^ fV sin^ 2ede = f C\l - cos 4 e)de TT B26 INTEGRAL CALCULUS The entire area of the four loops = ^^, which is half the area'of the circumscribed circle. 2. Find the entire area of the circle (Art. 135) r = a sin 0. Ans. ^. 4 In the two following curves find the area described by the radius vector in moving from 6 = to 6 = -. 3. r = sec 5 + tan (9. Ans. 1 + V2--. 8 4. r = a(l-tan2e). Ans. (^--V'- 5. Find the entire area of the cardioid (Art. 141) r = a(l — cos &). Ans. ^ , or six times the area of the generating circle. Also find the area from fl = - to (9 = ^. Ans. (3 tt — 2)— 4 4 '^ 8. 6. Find the area described by the radius vector in the parabola r = asec2^, from 61 = to e = -. Ans. ~ 2 2 3 Also find the area from = ^ to 6 = ^. Ans. ^"^ "^ 9V3' 7. Find the entire area of the lemniscate (Art. 143) r' = d' cos 2 6. Ans. aK 8. Show that the area bounded by any two radii vectores of the reciprocal spiral (Art. 137) r$=a is proportional to the difference between the lengths of these radii. 9. In the spiral of Archimedes (Art. 136), r = aO, find the area described by the radius vector in one entire revolution from 9 = Ans. il^. 2 Also find the area of the strip added by the nth revolution. Ans. A^sEst^sSa*. APPLICATION OF INTEGRATION TO PLANE CURVES 327 10. Find the area of the part of the circle r = asin$ +bcos 6, from 6 = 0tod = -. 2 Ans. "(«' + ^')+^. . 8 2 11. Find the area common to the two circles r = a sin 6 + b cos 6, r = a cos 6 + h sin 6. Ans. f^ + 2 tan-' ^ ^^!±A' - '^^^, where a>h. \2 aj 4: 4 12. Find the area of the loop of the Folium of Descartes (Art. 127) 3 a tan 6 sec 6 Ans. So" 1 + tan^ e 13. Show that the line r= -«secg (x + y=2a), divides the 1 + tane' ^ ^ ' area of the loop of the preceding example in the ratio 2 : 1. 14. Find the entire area within the curve (Art. 146) r = a sin' -, no o part being counted twice. Ans. (10:r + 9V3) ^. 249. Lengths of Curves. Rectangular Coordinates. To find the Length of the Arc PQ between Two Given Points P and Q. Let OA='a, OB = h. Denoting the required length of arc by s, we have from (1), Art. 156, ds-. ■■M% dx. Hence S-- and between the given limits B X the limits being the initial and final values of x. 328 INTEGRAL CALCULUS We may also use the formula the limits being the initial and final values of y, g=OG, and h=OH. EXAMPLES 1. Find the length of the arc of the parabola 2/^ = 4 ax, from the vertex to the extremity of the latus rectum. Here ? = 4 ax ^i therefore s= f'fl + lfdx^ Cf^+^Ydx. ,/0 \ xj Jit \ X J This may be integrated by Ex. 13, p. 306, making 6 = 0. rfa±x)\ ^^ ^ ^ax + x" + a log iVa^+x + Vx) rfa + xsj ^^^^ |-^2 + log (1 + V2)] =2.^558 a. Or we may use the formula (2), a-^lL^ dx^y_ 4 a' dy 2 a • = ^ [l^f + ^' + —- log {y + V2/^ + 4 aO = a[V2 + log(l+ V2)] APPLICATION OF INTEGRATION TO PLANE CURVES 329 2. Find the length of the arc of the semicubical parabola (Art. 130) ay^ = a?, from x = — to a; = 5 a. Ans. ——. 4 8 3. Find the entire length of the arc of the hypocycloid (Art. 132) x' + y^ = a Ans. 6 a. 4. Find the length of the arc of the catenary (Art. 128) from a; = to the point (x, y). Ans. g (e" - e "). y = log sec X, from a; = to a; = -. 5. Find the length of the arc of the curve Ans. log (2 + V3). 6. Find the length of the curve 17 6 a;2/ = a;* + 3, from a; = 1 to a; = 2. Ans. —. 7. Find the perimeter of the loop of the curve ^ay^={x-2a){x-5 af. Ans. 4 VSa. 8. Find the length of that part of the evolute of the parabola (Art. 167) 27 ay^ = 4t(x — 2 af included within the parabola y^j= 4 ax. Ans. 4(3V3-l)a. 9. Find the length of the curve y = log , from a; = 1 to a; = 2. e'^ + l Ans. log(e + 0- x\^ , fy)" 10. Find the length of one quadrant of the curve f - j + f ^ ) = 1 ■ a^ + ab + W Ans. a + & 11. The parametric equations of acurve are a; = e* sin 0, y~= e' cos 6. Find the length of arc from e = to 6 = |. Ans. V2 (e? - 1). 330 INTEGRAL CALCULUS 12. The parametric equations of the epicycloid, the radius of the fixed circle being a, and that of the rolling circle ^, are rc = - (3 cos ^ — cos 3 <^), 2/ = |(3sin^-sin3), I ^ being the angle of the fixed circle, over which the small circle has rolled. Find the entire length of the curve. Ans. 12 a. 250. Lengths of Curves. Polar Coordinates. To find the Length of the Arc PQ between Two Given Points P and Q. Let POX = a, QOX = p. We have from (3), Art. 156, ds = Mm '« therefore s = i'MI 0"] dO, (1) the limits being the limiting values q' of e. Or we have ds = 1 + r' therefore '-£[ l + r' dr ; (2), Art. 156, dr, ... (2) the limits being the limiting values of r. That is, OP = a, OQ = b. EXAMPLES 1. Find the length- of the arc of the spiral of Archimedes (Art. 136), r = ad, from the origin to the end of the first revolution. Here — = a, and we have by (1), APPLICATION OF INTEGRATION TO PLANE CURVES 331 = a[^4±i! + |log(. + VrH r Vl + 4,r2 +ilog (2 ;r 4- Vl + 4,rO Or we may use the formula (2) a=j%'i+,YfY^;. *■ 6=1 ^=1 a' d/- a Jr*1-na * a'' a Jo ct 1 ^ J dr 2n-a D 7rV4,r^ + l+^log(2,r + V4,r^ + 1) 2. Find the entire length of the circle (Art. 136) r = 2asixi.6. Ans. 2 ira. 3. Find the length of the arc of the circle r = a sin 6 + 6 cos d, from 6 = to (r, 6). Ans. VaF+V 6. 4. Find the length of the logarithmic spiral (Art. 138) r = e"* from ■the point {r^, Oi) to (rs, O^), using the formula (2), and the equation 6 = l2gi:. Ans. ^«^ + !(.,-,•). 332 INTEGRAL CALCULUS 5. Find tlie entire length of the cardioid (Art. 141) r =a (1 — cos &). Ans. 8 a. Also show that the arc of the upper half of the curve is bisected by 3 6. Solve Ex. 5 by using formula (2) and the equation = vers"'-. 7. Find the arc of the reciprocal spiral (Art. 137) r$ = a, from 8. Find the arc of the parabola (Art. 139) r = a sec^ - from = Otoe = 5. ^Ms. fsec- + logtani^k. 2 \ 4 ^ 8 J 9. Find the entire length of the arc of the curve (Art. 145) ■ y P ^3 TrCC r = a sm^ -. Ans. -^. Also show that the arc AB is one third of OABG. Hence OA, AB, BG are in arithmetical progression. 10. Find the entire length of the curve r = a sin" -, n being a posi- tive integer. '* See for integration Exs. 24, 26, p. 314. . „ 2-4-6-"?i o v ■Ans. - — — — — 2 a, when n is even. 1-0-5 ■■■ {n — 1) 1 • 3 ■ 5 • • ■ n , . , , - -rra, when n is odd. 2.4.6."(n-l) APPLICATION OP INTEGRATION TO PLANE CURVES 333 251. Volumes of Surfaces of Revolution. To find the Volume gener- ated by revolving about OX the Plane Area APQB. Let OA = a, OB = b. Let X and y be the coordinates of any point P2 of the given curve. It is evident that the rectan- gle P2A2A3 will generate a right cylinder, whose volume is iry^ Ax. The sum of all these cylinders may be represented by ^f a Ax. Ai As As Ai B X The required volume is the limit of the sum of the cylinders, as Ao; approaches zero. That is, Fi = 7r I y^dx. Similarly the volume generated by revolving PGHQ about F is Vy = -rr \ x^ dy, where 0G = g, and OH = h. EXAMPLES 1. Find the volume generated by revolving the ellipse about its major axis, OX. This is called the prolate spheroid. V 2 = .Jj^dx = .j^ -ia^-x^dx = -^(a^x--^ = 2wab'^ F=|7ra&2. 334 INTEGRAL CALCULUS 2. Find the volume generated by revolving the ellipse about its minor axis, OY. This is called the ohlate spheroid. V 2' -'£^''''='^1^'-^'^^^= 2 7ra% V=-7ra'b. 3 3. If the parabola y'^ = ^ax is revolved about OX, show that the volume from a; = 0toa; = 2ais one third the volume from x = 2a to a; = 4 a. 4. Find the volume generated by revolving the segment LOL' of the parabola about the latus rectum LL'. Here -^ = tt ) {PNfdy = tt (a - xfdy Ans. —- tto?. 15 5. Find the volume generated by revolving about OX one loop of the curve (Art. 134) a*2/^ = d?x'^ — a^. Ans. Trpira^ 35 Y d/ L N 2a a F X 6. Find the entire volume generated by revolving about OX the hypocycloid (Art. 132) x^ + y^ = a^. 32 Ans. —^ tto?. 105 7. Find the volumes generated by revolving about OX, and about OY, the curve (Art. 133) ("^ + (^^=.1. Ans. V^ = — ivah 5 APPLICATION OF INTEGRATION TO PLANE CURVES 335 8. The part of the line - + ^ = 1, intercepted between the coor- a dinate axes, is revolved about the line x = 2a. Find the included volume. Ans. '^■n-a^b. 9. The segment of the parabola, x' — 3x + 2y = 0, above OX, is D-| revolved about OX. Find the volume generated. Ans. — - ■ 10. A segment of a circle is revolved about a diameter parallel to its chord. Show that the volume generated is equal to that of a sphere whose diameter is equal to the chord. 11. Find the volume generated by revolving about OF the witch (Art. 126), y = ^"^ ^ , from (0, 2a)toy = a. Ans. 4 (log 4-1) W. 12. Find the volume generated by revolving the upper half, ABA', of the curve (Art. 133) /"-Y + f ^Y = 1, about the tangent at B. 13. Find the volume generated by revolving about OX the area included between the ellipse — + --^ = 1, and the parabola 2 ay'^ = 3 Wx. a- b' -|^g Ans. — TTob". 48 X X 14. A segment of the catenary (Art. 128), y = -[e'' + e "), by a chord through the points x = ± a log 2, is revolved about the tangent at the vertex. Find the volume generated. Ans. 3flog2-iij7ra^ 15. Find the volume generated by revolving about the latus rec- tum of the ellipse ^ + ^ = 1, the segment cut off by the latus Of / M h'' rectum. Ans. 2irlab^—- aSVa'' — 6^sin~'- V 3a a 336 INTEGRAL CALCULUS 252. Derivative of Area of Surface of Revolution. In order to obtain the formula for the surface generated by the revolution of a given arc, it is necessary to find the derivative of this surface with respect to the arc. Let S denote the surface gen- erated by revolving about OX the arc s, AP. Using for abbreviation the expression " Surf ( ) " to denote "the surface generated by re- volving ( ) about OX," we have S = Surf (s), A^ = Surf (As). This may be written A[l + (|- t dx. Or we may use (■fo= 1 + f — and instead of (1) we have S^ and instead of (2) ^„ = 2.f.[l+(| dy. 1 dy. B X (2) (1') (2') 338 INTEGRAL CALCULUS EXAMPLES 1. Find the area of the surface generated by revolving about OX the hypocycloid (Art. 132) x^ + y^ = a^- Here («/ = a^ — a;')^, -^= — (a' —xi)ix ^_ Using (1) ls^^2.C\al-J)^h + '-^'t'^^ i Jo L a;* J = 2 ^ r{ai - xt)l ^ dx = 2 irai fCa^ - x^fx'^ Jo X3 Jo dx 6 TTO' „ 12 TTtt^ 3 '5, = Or we may use (1') — = — (a^ — yi)^'y i- I ^. = 2 ^£yh + '^^^1'% = 2 ^a^J^ V dy 6 7ra^ y 2. Show that the area of the surface generated by revolving the parabola 2/^=4 ax, about OX, from a; = to x = 3 a, is one eighth of that from a; = 3 a to a; = 15 a. 3. Find the area of the surface generated by revolving about OX the loop of the curve 9ay^ = x{3 a— xf. Ans. 3 tto?. 4. Find the surface generated by revolving about OX, the arc of the curve 6 a^ a;?/ = a;^ + 3 a*, from a; = a to a; = 2 a. Ans. — Tral 16 5. The arc of the preceding curve from a; = a to a; = 3 a, revolves about Y. What is the surface generated ? Ans. (20 + log 3)7ral 6. Find the surface generated by revolving about OFthe curve 4 2/ = x^ — 2 log X, from a; = 1 to a? = 4. Ans. 24 -n-. APPLICATION OF INTEGRATION TO PLANE CURVES 339 7. Find the entire surface generated by revolving about OX the ellipse 3 a^ + 4 2/2 = 3 a^. Ans.(^ + ^ 2 V3J ^ 8. Find the entire surface generated by revolving about OY the preceding ellipse. ^„^_ (4 + 3 log 3)^1 9. Find the surface generated by revolving about OX the loop of the curve 8 aV = aV — a;*. . tto' " Ans. 4 10. An arc, subtending an angle 2 a, of a circle whose radius is a, revolves about its chord. Find the surface generated. Ans. 4 7ra2 (sina — aCOS a). 11. The are of the catenary (Art. 121) y = -fe^ + e~«\ from x = a to a; = 2 a, revolves about Y. Find the surface generated. Ans. (e^ + 2e-'-3e-^7ra\ 12. The parametric equations of a curve are a; = e* sin 9, y = e^ cos 9. Find the surface generated by revolving the arc from 9 = to e = \ about OX. Ans. ^(e''-2). 13. Find the surface generated by revolving about Y the arc of the preceding example. ^^g ^ /2 e" + 1") o 14. The parametric equations of the epicycloid, the radius of the fixed circle being a, and that of the rolling circle - (Ex. 12, p. 330) are x = — cos <^ — -cos 3, y = ■— sm (/> — - sm 3 . ^ Zi Zi z Find the entire surface generated by revolving the curve about OX. Ans. -irV. 15. Find the surface generated by revolving one arch of the pre- ceding curve about Y. Ans. 6 W. 340 INTEGRAL CALCULUS 254. Volume by Area of Section. The volume of a solid may be found by a single integration, when the area of a section can be ex- pressed in terms of its per- pendicular distance from a fixed point. Let us denote this dis- tance by X, and the area of the section, supposed to be a function of x, by X. The volume included between two sections sep- arated by the distance dx will ultimately be Xdx, and we have for the volume of the solid the limits being the initial, and final, values of x. EXAMPLES 1. rind the volume of a pyramid or cone having any base. Let A be the area of the base, and 7i the altitude. Let X denote the perpendicular distance from the vertex of a sec- tion parallel to the base. Calling the area of this section X, we have, by solid geometry, X A' X = Ax" Hence, J"" A C Xdx = ~ \ a? dx = h Jfi Ah!" h'3 Ah 3 ' 2. Find the volume of a right conoid with circular base, the radius of base being a, and altitude h. 0A = BC=2a, BO=OA = h. The section MTQ, perpendicular to OA, is an isosceles triangle. APPLICATION OF INTEGRATION TO PLANE CURVES 341 Let X = OP; then X = area RTQ, = PTxPQ = h ■v'2 ax — x'. Hence, V= I Xdx = h | V2aa; — ar' da; = ^ — This is one half the cylinder of the same base and altitude. 3. Find the volume of the ellipsoid _ 9 *^ -L^.^^ .9 (1) Let us find the area of a section C'B'D' perpendicular to OX, at the dis- tance from the origin OM=x. This section is an ellipse whose semi- axes are MB' and MC. To find MB', let z = in (1), and we have 'f J "' \ oi/ U \/\ ^<\ a 7/ i I / 1 "' I , = MB' = --s/W^^. a To find JfC", let 2/ = in (1), and we have z = MC = --Va'-x'. a The area of the ellipse (Ex. 2, p. 321) is ^(MB') {MO). Hence, rbc X='^{a?-«?), and V=2£xdx = ^/° {a?-^dx = ^ irabc. 342 INTEGRAL CALCULUS 4. A rectangle moves from a fixed point, one side varying as the distance from the 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 ? Ans. 4i cubic feet. 5. The axes of two right circular cylinders having equal bases, radius a, intersect at right angles. Tind the volume common to the two. -J 16 a^ Ans. -— — 6. A torus is generated by a circle, radius b, revolving about an axis in its plane, a being the distance of the centre of the circle from the axis. Pind the volume by means of sections perpendicular to the axis. Ans. 2 T!^a-b. 7. A football is 16 inches long, and a plane section containing a seam of the cover is an ellipse 8 inches broad. Find the volume of the ball, assuming that the leather is so stiff that every plane cross- section is a square. Ans. 341-J^ cu. in. 8. Given a right cylinder, altitude h, and radius of base a. Through a diameter of the upper base two planes are passed, touch- ing the lower base on opposite sides. Find the volume included between the planes. . f _^\ n ns. ^n -ja I. 9. Two cylinders of equal altitude h have a circle of radius a, for their common upper base. Their lower bases are tangent to each other. Find the volume common to the two cylinders. Ans. i^. CHAPTER XXIX SUCCESSIVE INTEGRATION 255- Definite Double Integral. — A double integral is the integral of an integral. Thus, X and y being independent variables, the definite double integral, ^„ ^, I I /(a;, y)dx dy, indicates the following operations: Treating a; as a constant, integrate f{x, y) with respect to y between the limits d and c ; then integrate the result with respect to X between the limits 6 and a.* Por example, J ':^(}i — y)dxdy=\ x'lhy — •J-\dx= \ a?~dx _ 6^ ^ 1 2- _ 7 a^V ~2 3|„ ~ 6 Notice that the order of the two integrations is indicated in the given definite integral by the order of the differentials dx dy, taken I I being used in the same a t/o order. It should be said, however, that the order of the integrations is denoted differently by diiferent writers. 256. Variable Limits. — The limits of the first integration, instead of being constants, are often functions of the variable of the second integration. * Using parentheses, this mightbe represented by ( ( j f{x,y)dy\dx. 343 344 INTEGRAL CALCULUS For example, jT'jf " xy dy "^^ = jrY|T"_J/ dy = \£(? 2/^ + 2 ay' - a'y)dy = ^^ As another example, J J '"~''\x + y)dxdy=j fxy + Yj^'"'""dz When the limits are all constants, as in Art. 248, the order of the integrations may be reversed without affecting the result. That is, I a^(6 — y)dx dy=i I ar(b — y)dy dx. Where the definite integral has variable limits, the order of integra- tions can be changed only by new limits adapted to the new order. 257. Triple Integrals. — A similar notation is used for three suc- cessive integrations. Thus III a^yhdxdydz= I I —-x'y^dxdy 2 Jb 3 2 \3 3j 6 ^ ' EXAMPLES Evaluate the following definite integrals : I xy\x— y) ax ay = 2. j ( 7^ sijx 6 drdd = '^^ — (cos y3 — cos a). Jb J a o nb ^2^2 xy{x-y)dxdy = —{a-b). SUCCESSIVE INTEGRATION 345 *• XX rdrde = ^-^- 5. I j r^sm6dedr = ^- nlOy V «!/ — y^dydx = 6 a^. 2 r^f 7*dBd —( -— ^— • ^0 J a COB e \ 15 J 10 7r n29 1 sm(2,j>-e)ded = ^- 10. J'"J"'jsiii2<^dfd<^ = ^' + |- Jo Ji2 a; +2/ 2 a 12. f f r"(a^ + f + z^ dx dy dz = ^ (aF + b' + c^. fc/o -A) •/} ^ 13.1)1 sin (xyz) dxdydz = -' I uvw du dv dw = -— • ^0 */u o CHAPTER XXX APPLICATIONS OF DOUBLE INTEGRATION 258. Moment of Inertia. If r^, r^, n, •••, r„ are the distances from a given line of n particles of masses m-i, irii, m^, • ■ •, m„, the sum Mil?-; '■ + m^r.? + m^ri + • ■ ■ + in„r^ = ^ {m)^ is defined in treatises on mechanics as the moment of inertia of the system about the given line. The moment of inertia of a continuous solid about a given line is the sum of the products obtained by multiplying the mass of each infinitesimal portion of the solid by the square of its distance from the given line. The summation is then effected by integration, and we have for the moment of inertia of a body of mass M, '= Ci^dM. 259. Moment of Inertia of a Plane Area. The moment of inertia of a given plane area about a given point may be defined as the sum of the products obtained, by multiplying the area of each infini- tesimal portion by the square of its distance from 0. This may be regarded as the moment of inertia of a thin plane sheet of uniform thickness and density, about a line through per- •pendicular to the plane, the mass of a square unit of the sheet being taken as unity. We shall illustrate double integration by finding the moment of inertia of certain areas. 346 APPLICATIONS OF DOUBLE INTEGRATION 347 B M' N' -4- — I ZOZl -i-- M N 260. Double Integration. Rectangular Coordinates. To find the moment of inertia of the rectangle OAGB about 0. Let OA = a, OB = h. Suppose the rectangle y divided into rectangular elements by lines parallel to the coordinate axes. Let X, y, which are to be re- garded as independent vari- ables, be the coordinates of any point of intersection as P, and X + dx, y + dy the coordinates of Q. Then the area of the element PQ is dx dy. Moment of inertia of PQ = OP •dxdy = {v? + y^) dx dy. The moment of inertia of the entire rectangle OAOB is the sum of all the terms obtained from (d^ + y'^) dx dy, by varying x from to a, and y from to h. If we suppose x to be constant, while y varies from to 6, we shall have the terms that constitute a vertical strip MNN'M'. Hence Moment of inertia of MNN'M' = dxC (9? + y^) dy ■.dx{x^y + Yj^= ha? 4- -\dx. Having thus found the moment of a vertical strip, we may sum all these strips by supposing x in this result to vary from to a. That is, the moment of inertia of OAGB, The preceding operations are those represented by the double integral, I=££{x' + f)dxdy. 348 INTEGRAL CALCULUS If we first collect all the elements in a horizontal strip, and then sum these horizontal strips, we have 1= C C{a? + y^dydx = a?h + aW 261. Variable Limits. To find the moment of inertia of the right triangle OAG about 0. Let OA = a, AO=h. The y ^C equation of OC is b y = -x. a This differs from the pre- ceding problem only in the limits of the first integration. In collecting the elements in a vertical strip MJSf, y varies from to MN. But MN is no longer a constant as in Art. 260, but varies with OM, according to the equation of OC, y = -x. Hence the limits of y are and -x. a In collecting all the vertical strips by the second integration, x varies from to a, as in Art. 260. Thus we have for the moment of inertia of OAO, ''So" So " ^"^ "*" ^'^ ^"^ ^y " "^ (l + ^ 12 By supposing the triangle composed of horizontal strips as HK, we shall find Y /= r C\o? + y')dydx -- 8. Find by a double integration the area included between the parabolas y^ = 3x, and y^ = 12 (60 — x). Arts. 960. 9. Find the moment of inertia of the area included between the parabola y' = iax, x = 4:a, and the axis of X, about the focus of the parabola. ^ . 2336. Ans. 35 10. Find the moment of inertia of the area included between the lines y'=2 X, x + 2y = 5 a, and the axis of X, about the intersection of the first two lines. Ans. 125 g^ 6 263. Double Integration. Polar Coordinates. To find the area of the quadrant of a circle AOB, whose radius is a. In rectangular coordinates, Art. 260, the lines of division consist of two systems, for one of which x is constant, and for the other, y is constant. So in polar coordinates, we have one system of straight lines through the origin, for each of which 6 is constant, and another system of circles about the origin as centre, for each of which r is constant. Let r, 6, which are to be regarded as independent variables, be the coordinates of any point of intersec- '-■h-J /^■^\ \ N N' A APPLICATIONS OF DOUBLE INTEGRATION 351 tion as P, and r + dr, 6 + dO, the coordinates of Q. Then the area of PQ is ultimately- Pi? X-RQ=rc?6l -dr. If we first integrate, regarding constant while r varies from to a, we collect all the elements in any sector MOM'. The second iategration sums all the sectors, by varying 6 from 2 Hence Area BOA ■■ Jo Jo r d& dr = 4 If we reverse the order of integration, integrating first with respect to 0, and afterwards with respect to r, we collect all the elements in a circular strip NLL'N', and sum all these strips. This is written Area BOA •/n ^0 rdri 264. If the moment of inertia about is required, we have for the moment of inertia of PQ, r^-rd6dr. Hence, the moment of BOA is /= r r^dddr= C" Ci^drde^^- Jo Jo Jo Jo o 265. Variable Limits. To find by a double integration the area of the semicircle OB J. with radius 0C=a, the pole being on the circumference. The polar equation of the circle is r = 2a cos 0. If we integrate first with respect to r, then with respect 'tcr$, we shall have Area OBA /»2 /»2( Jo Jo rd6dr=- Here, in collecting the elements in a radial strip OM, r varies from to OM. But OJf varies with 0, according to the equation of the circle r = 2 a cos 6. Hence the limits are and 2 a cos 6, 352 INTEGRAL CALCULUS In collecting all these radial strips for the second integration, 6 varies from to - • 2 By supposing the area composed of concentric circular strips about as LK, we find Area OBA «/o rdrdd=' EXAM PLES 1. Find the moment of inertia about the origin of the area in- cluded between the two circles, r = a sin 6 and r = b sin 6, where a > 6. Ans. 1= «/0 •/ 6 SI "• a ain Q ')^d9dr = -^7r(a'-b*). /Q •y 6 sine o2 2. Eind the moment of inertia about the origin of the area between the parabola (Art. 139), r = asec^-, its latus rectum, and 0^- Ans. ^^ 35 3. Find the moment of inertia about its centre of the area of the lemniscate (Art. 143) r' = d' cos 2 6. Ans. irO' 4. Find by double integration the entire area of the cardioid (Art. 141) r = a(l-cose). , ^^^ Sttci^ 5. Find the moment of inertia about the origin of the area of the preceding cardioid. . 35'n-a'^ 16 APPLICATIONS OF DOUBLE INTEGRATION 353 6. Mud the moment of inertia about its centre of tlie entire «rc Ans. of the four-leaved rose (Art. 144) r = a sin 2 6. . Zva*^ 16 7. Find by a double integration the area of one loop of the lemniseate (Art. 143) outside the circle 2 r^ = al , f /'^ '"\ ^ 8. Find the moment of inertia of the area of the preceding example about the centre of 'the lemniseate. . / V3 tN a* "*' (,^"*'3Jl6' 266. Volumes and Surfaces of Revolution. Polar CoSrdinates. If in the figure of Art. 263 we suppose a revolution about OX, the volume generated by the infinitesimal area PQ is the product of this area by the circumference through which it revolves, that is, 2 ■n- r sin -rdQ dr. Hence for the entire volume V= 2 TT C Ci^ sin Odd dr, the limits being determined as in Art. 263. If the revolution is about Y, V=-27r C Ci^ cos Ode dr. 'IS' The area of the surface generated about OX is (Art. 253) S = 27rCyds = 2 7rCrsme .fdr ^n^;^' dd. EXAMPLES 1. Find the volume generated by revolving the cardioid (Art. 141) »• = a (1 — cos 6) about OX , 8 , , . ,, . ., , , ^ ^ Ans. - ira^, tvnce the inscribed sphere. o 2. Show that the entire volume generated by revolving the four- 1 (\ leaved rose (Art. 144) ?•= a sin 2 6 about OX is — of the volume of the circumscribed sphere. 354 INTEGRAL CALCULUS 3. Find the volume generated by revolving one loop of the four- leaved rose r = a sin 2 6 about the axis of the loop. Ans. (sV-2- 9)^. 4. Find the volume generated by revolving the lemniscate (Art. 143) ?-2 = a' cos 2 about Y. ^'- a" V2 Ans. „ ■ 5. Find the volume generated by revolving the lemniscate about OX. ^a? Ans. —— ^log(V2 + l)-i .V2 ^. 6. Find area of surface generated by revolving the cardioid r=a(l — cos6) about OX. . 32Tra^ 7. Find the moment of inertia of a sphere (radius a) about a diameter, m being the mass of a unit of volume. Ans. S^n-g^m 15 CHAPTER XXXI SURFACE, VOLUME, AND MOMENT OF INERTIA OF ANY SOLID 267. To find the Area of Any Surface, whose Equation is given between Three Rectangular Coordinates, x, y, z. Let this equation be „/ , z =f(x, y). Suppose the given surface to be divided into elements by two series of planes, parallel respectively to XZ and YZ. These planes will also divide the plane XY into elementary rectangles, one of which is PQ, the projection upon the plane XY of the corresponding element of the surface P'Q'. Let X, y, z be the coordinates of P' and x + dx, y + dy, z + dz, of Q'. 355 356 INTEGRAL CALCULUS Since PQ is the projection of P'Q,', the area of PQ is equal to that of PQ' multiplied by the cosine of the inclination of PQ' to the plane XT. This angle is evidently that made by the tangent plane at P with the plane XY. Denoting this angle by y, Area PQ = Area PQ' ■ cos y, Area PQ' = Area PQ • sec y. We see from the figure that Area PQ = dx dy. Also from (8), Art. 110, sec y = 'i+m'+f'j- \dxj \dyj J ■where — and — are partial derivatives, taken from the equation of dx dy ^ ' ^ the given surface z =f(x, y). Hence Area PQ' = fl + [^Y + [— Y dxdy. If S denote the required surface, S =/iI-(l)'-(|, dxdy, (1) the limits of the integration depending upon the projection, on the plane XY, of the surface required. Tor example, suppose the surface ABC to be one eighth of the surface of a sphere whose equation is ii? + y^ + z^ = a^. Here dz__ dx X dz _ z' dy' 1 + (£)'HI)"--^ ' d' — x' — '( SURFACE, VOLUME, AND MOMENT OF INERTIA 357 Substituting in (1), we have S J J y/w dxdy This is to be integrated over the region OB A, the projection of the required surface on the plane XY. The equation of the boundary AB is a? -\-'jf = a?. Integrating first with respect to y, we collect all the elements in a strip M'N'KL, y varying from zero to ML, that is, between the limits and Va^ — 3?. Integrating afterwards with respect to x, we sum all the strips, to obtain the required surface ABC, x varying from to a. Hence S Jo Jo ■v^aS— x2 dxdy ■\/a? — a? — y' '' 2 Another example is the following: The centre of a sphere, whose radius is a, is on the surface of a right circular cylinder, the radius of whose ft base is rind the surface of the sphere intercepted by the cylinder. Take for the equa- tions of the sphere and cylinder, a^ + 2/' + «' = «', and oi? + y'' = ax. CPAQ is ope fourth the required surface. Since this surface is a part of the sphere, the 358 INTEGRAL CALCULUS partial derivatives ~, — must be taken from ap + y'^ + z^ == a^. ax ay giving, as in the preceding example, S dxdy Va' to be integrated over the region OB A, the projection of CPAQ on the plane XY. The equation of the curve ORA is x^ + y^ = ax. Hence ^S= C" C"^' adxdy_ ^(._^^^, 4 Jo Jo ^ci'-x' — v'^ Let us now find the surface of the cylinder intercepted by the sphere, one fourth of which is GPARO. Since this is a part of the cylinder x'' + ^ = ax, the partial derivatives in (1) must be taken from this equation. But from 3^ -\-y'':=ax, we &[id. — = oo, — = oo- 9a; dy The formula (1) is, then, inapplicable in this case. It is also evident from the figure that the surface CPARO cannot be found from its projection on the plane XY, since this projection is the curve ORA. The difficulty is removed by projecting on the plane XZ, and using, instead of (1), We now find from af + y^ = ax, dy ^ a-2x Sy^^ dx y ' dz Substituting in (2), and simplifying, 1 o _ r r a dx dz 4 J J 2 -Vax-x^ SURFACE, VOLUME, AND MOMENT OF INERTIA 359 This must be integrated over the region CP'AO, OP' A being the projection on XZ of CPA. To find the equation of CPA, we eliminate y from x' + y^ + z' = a^ and x^ + y'' = ax, giving, z^ = a' — ax. Hence l^_a p r "'"""'" ^ - X' dx^ ^-zax-x' + a vers-' ^ • J a; a 384 INTEGRAL CALCULUS CO r ^2ax — x^dx _ _ (2 ax — af)'^ J 3? ~ 3ai? ' 53./ '" 54. / (2 ax — x-)^ dr^2 ax — of xdx X 55. (2 ax — af)i a V2 ax — of INTEGRALS CONTAINING ±ax^+bx + c dx _ 2 _i 2 aa; + & — ; — ; = — =^^= tan' ax' + bx + c V4 ac - b"- ■\/4 ac — &^ V4 ac — 6^ 56. or ^ ^^^2aa; + 6-V6^-4ac 57. V&^ — 4 oc 2 ax + 6 + V^^ — 4 ac /da; 1 ,_ , —===== = — = log (2 ax + 6 + 2VaVaa? + &K + ?). V aor + 6a; + c V a 58. rVaa;2^ja; + cda;=?-!^^-±-^Vaa;2+&a; + c •^ 4a ~ "" log (2 aa; + & + 2VaVaa;^ + &a; + c). 8a^ 59. f ^'^ ^-Lsin- ^^IlT-L . •^ V — aa;^ + 6a; + c Va V6^ + -1 ac 60. j V— a*^ + 6a; + c da; 2 aa; — 6 / — „ » , , — ; — ,6^ + 4 ac ■ _, 2ax — b = V — ax' + ox-\-c-\ sm ' — . * a 8 a* V¥ + 4 ac INTEGRALS FOR REFERENCE 385 OTHER INTEGRALS "■ /VS;* = V(a + x){b + x) + (a — b) log ( Va + a; + V6 + x). 62. I a/- — -dx = V(a — a;)(& + a;) + (a + 6) sin~'-i/— + & INDEX PAGE Acceleration . . 18 Angles, between two curves . . 174, 183 with coordinate planes . 133 Arc, derivative of 186, 187 Area, any surface ... . . 355 derivative of 241 of curve . . 242, 306, 320, 325, 349 surface of revolution . 337, 353 Asymptotes 179 Attraction at a point 375 Cauchy's test for convergence . . 82 Centre, of curvature . . 200 of gravity ... . 365 of pressure 373 Change of variable . . .57, 58, 148, 263, 299, 304, 317 Circle, of curvature . 195, 200 osculating 209 Comparison test for convergence . 80 Computation, by logarithms . 94 ofT 96 Constant, definition of . . .1 derivative of . . 26 notation of ... . . 1 of integration . . . 224, 315 Contact, order of ... . 206, 207, 209 Convergence, absolute and condi- tional 78 interval of . 86 tests for ... .79 Curvature, centre of 200 circle of ... . 195, 200 direction of ... 189 radius of . . 193, 196, 197 uniform . . 193 variable .... 194 Curves, angle of intersection of . . 197 area of . . 242, 306, 320, 325, 349 continuous and discontinu- ous 22 PAGE Curves, direction of . . . 16, 174, 182 for reference, higher plane . 162 length of 327, 330 osculating . . . 208 Definite integrals . . . 307 as a sum . . 319 definition of 310 double . . 343 sign of . . 314 Differential coefficient . . .12 Differentials, definition of . . . 68, 70 formulae 71 Differentiation, algebraic formulas 26, 29 definition of . .26 inverse trigonomet- ric formulae . 51 logarithmic and ex- ponential formu- la; .. . .39 order of . . . 137 partial .... i:JO successive .... 61 trigonometric formu- la ... .45 Derivative, definition 11 general expression of . 12 illustrations of ... . 13 meanings of . . . 16, 17, 18 of an arc 186, 187 of area . . 241, 336 of function of a func- tion . ... 58 partial ... . 130 partial, of higher order 13G, 137 relation between — clx and liil 57 (iy total ... . . 140 INDEX 387 Element, of area 320 of an integral 319 Envelopes, definition of . 214, 215 equation of 215 of normals . . 217 Equation, of envelopes 215 of evolute 201 of normal . . 133 of tangent 133 parametric ... . . 324 Evolute 201 an envelope 217 equation of 201 properties of . . 202, 204 Function, algebraic 2 continuous . . .22, 315 definition . . ... 1 discontinuous . . .22 expansion of . . .88 implicit, differentiation of 75, 144 increasing and decreasing 21 inverse trigonometric . . 51 logarithmic 39 of two or more variables 156 transcendental .... 2 trigonometric ... .45 ... .365 Integration, containing (aa!+6)« . containing s r {ax +b) «, (ax + 6)» containing Gravity, centre of Higher plane curves 162 Huyghens's approximate length of arc 93 Increment . 11 Indefinite integral 309 Indeterminant forms . ... 106 Inertia, moment of . ... 346, 363 Infinite, limits 315 variables . . ... 315 Infinitesimals, order of 73 Inflexion • 190 Integration and derivative of integral 270 as a summation .... 306 between limits .... 309 by algebraic substitution 299 by parts . . -279 by substitution . 218-226 constant of . 224, 310 containing ^^i^::^, V~^±a^ . 296 263 264 Vi x2 + aa; + 6 . . . 266 definite 307 definition of 22.'i double . 343, 347, 349, 35(: evaluation of .... 30'i for reference .... 378 fundamental 225 indefinite . . ... 309 of sec" X OiX, cosec x dx . 274 of sin" a; dx, COS" a; da; . 270 of sin"* X cos" X dx 271, 276, 291, 293 of tan" a; da;, cot" a; da; . 273 of tan" X sec" x dx, cot" X cot" X dx 274, 291, 293 of e"" sin nx dx, e"* cos nx dx of a;'" (a + 6a;")*' da; . . of/(a;2)a;da; . . . of rational fractions . proofs of formulae . . successive triple . Intercepts of tangent Involute . . • properties of 283 284 295 249 227 343 344, 361, 363 . . 173 ... 201 202, 204 Leibnitz's theorem 65 Length of curves .... 93,327,330 Limit, change of 317 definition of 8 infinite 315 Napierian base . . . . 10 notation of 8 relation of arc to chord . . 9 variable ... . 343, 348, 351 Liquids, pressure of 370 Logarithmic functions ... 2, 39 Logarithms, computation by . . • 94 Napierian 41 Maolaurin's theorem .... 89 Maxima and minima .... 114, 155 Moment of inertia 346, 363 388 INDEX PAGK Napierian base 8 Normals 133 Order, of contact 206 of differentiation 137 of integration 343 Osculating curves 208 order of contact of . . 209 Osculating circle, coordinates of centre 209 radius of 209 Pappus, theorems of 369 Parameter 214, 324 Parametric equations 324 Power series 85 Pressure, centre of 373 of liquids 370 Rates 18 Reduction formulse 284, 291 Remainder, Taylor's theorem . . . 105 Series, computation by 94 convergence of power ... 85 convergent and divergent . . 78 of positive and negative terms 78 power 85 Slope of a curve 16 of a line 16 of a plane 133 Subtangent 173, 183 Subnormal 173,183 Surfaces, area of any 355 Surfaces of revolution, areas of 337, 353 derivative of area of . . 336 volumes of . 333 Tangent 70 intercept of 173 Tangent planes 133 Taylor's theorem .... 97, 103, 145 Theorem, Leibnitz's 65 Maclaurin's 89 mean value .... 84, 86 Pappus's 369 RoUe's , 83 Taylor's ... 97, 103, 145 Transformation 152, 153 Trigonometric functions ... 2, 45 Uniform curvature ...... 193 Unit of force ........ 376 Variable, change of . . . 57, 58, 148, 263, 299,304, 317 curvature 194 definition of 1 dependent 2 independent 2 notation of 1 Velocity 17, 18 Volumes, any solid 361 by area of section . . . 340 surfaces of revolution 333, 353 Wmm^r mmm I ! 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