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Do not deface books by marks and writing. Cornell University Library arV19577 ' Elements of the differential and Integra 3 1924 031 261 815 olin.anx The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31 924031 261 81 5 ELEMENTS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS CAMBRIDGE UNIVERSITY PRESS Unnlon: FETTEE LANE, E.G. C. F. CLAT, Manaoer (iniinSutrt: 100, PRINCES STREET 36«Iin: A. ASHER AND CO. leipjis: P. A. BROCKHAUS iJefa Sorfi: G. P. PUTNAM'S SONS Bomftag ttnlr Calcutta: MACMILLAN AND CO., Ltd. lAll rights reserved] ELEMENTS OF THE DIFFERENTIAL AND INTEGRAL CALCULUS By A. E. H. LOVE, M.A., D.Sc, F.R.S. Sedleian Professor of Natural Philosophy in the University of Oxford. Honorary Fellow of Queen's College, Oxford, Formerly Fellow and Lecturer of St John's College, Cambridge Cambridge : At the University Press 1909 eDambrilige : PRINTED BY JOHN OLAY, M.A. AT THE CNIVEESITY PEE3S. PREFACE IN the last six years I have given annually a course of about twenty lectures on the Elements of the Differential and Integral Calculus to classes consisting chiefly of students of Chemistry and Engineering. The work of preparing and delivering such lectures, and of revising them from year to year, teaches the lecturer many things in regard to the nature of the difficulties which are encountered by students. He is led to depart frequently from the traditional order of the subject matter, and to devise numerous simplifications in the proofs of propositions. It soon appeared that the amount of mathematical knowledge which need be possessed by a student before attempting the Calculus is very much less than has been supposed. For example, the Binomial Theorem in Algebra and the Addition Equation in Trigonometry are quite unnecessary. This book is written with the view of making the subject more easily and generally accessible than it has been hitherto. The principles of the Differential and -Integral Calculus ought to be counted as a part of the intellectual heritage of every educated man or woman in the twentieth century, no less than the Copemican system or the Darwinian theory. In order to make a beginning no previous knowledge of mathematics is needed beyond the most elementary notions of geometry, a little algebra. VI PREFACE including the law of indices, and the definitions of the trigonometric functions. In order to advance very far in the subject a student must advance in other branches of mathematics as well. This book is intended merely to help the reader to make a beginning. In order to render his progress as easy as possible, results with which he is supposed to be more or less familiar are recapitulated in the places where they are wanted, and formal proofs of some propositions are omitted from the text and placed in Appendices, along with certain rather abstract discussions. Two things in the subject are, and apparently must continue to be, difficult. These are the actual integration of particular functions and the theory of the exponential function. For a reader in search of culture the practice of integration is not very important, but for a student who wishes to make use of the Calculus it is indispensable. The difficulty seems to be purely one of technique. The best that I can do to meet it is to lead up gradually to the appropriate methods, and to illustrate them by sufficiently numerous and sufficiently easy examples. The student must not allow himself to be discouraged too easily by a few failures. The theory of the exponential function, on the other hand, is essentially difficult, and the history of mathematics shows what a formidable stumbling-block it proved itself from the time of the invention of logarithms by John Napier to the time of the revision of the foundations of mathematical analysis by Augustin Louis Cauchy. For the purpose of a study of the elements of the Calculus the whole of the theory is not required, for example, the exponential theorem is unnecessary, but it is necessary either to prove or to assume at least one PREFACE Vll proposition the rigorous proof of which is difficult. My plan has been to assume the existence of the exponential limit. This limit presents itself naturally in the process of diffe- rentiating a logarithm, while logarithms arise naturally, though -not historically, from the use of indices of powers. It would be unsatisfactory to' assume the existence of the limit without explanation, and for this reason an arithmetical argument has been given which makes it appear probable that the limit exists. Such arguments are often more convincing than formal proofs. It would be unsatisfactory to substitute such an argument for a proof, and merely irritating to refer the reader to a proof in some book which he may not possess, and a formal proof is given in an Appendix (pp. 191 — 194). I have not tried to select one which is brief, or easy to reproduce in examinations, but my choice was guided by the wish to use none but the simplest mathematical material. The student who wishes to master the subject within the range of this book is recommended to work the Examples. In working some of these, a book of Tables of mathematical functions is needed, and it is assumed that the reader knows how to compute by means of logarithms. This knowledge is not, however, required in order to read the greater part of the text. My best thanks are due to Mr F. B. Pidduck, Fellow of Queen's College, Oxford, for his kindness in reading and correcting the proofs. A. E. H. LOVE. Oxford, July 1909. CONTENTS CHAPTER I INTRODUCTORY iBT. PAGE I Examples of relations between the measures of variable quantities 1 2 — 5 Graphic representation of such relations . . 2 6 Uniform motion .... . . 6 7 — 8 Gradient of a rectilinear graph .... 7 Examples 9 9 Graphic representation of the motion of a falling body 9 Examples 10 10 Velocity of falling body 10 II Gradient of a parabola 11 12 Generalization . . 12 Examples ... . . 13 Additional Exercises ... . . 14 CHAPTER II DIFFERENTIATION 13 Meaning of "function" . 16 14^17 Natiire of the process of differentiation 16 18 Uses of the process 18 Examples . 21 19,20 Meaning of "limit" 21 21 Rules of differentiation . 23 22,23 Differentiation of integral powers and rational integral functions 25 Examples . 27 X CONTENTS ABT. PAOB 24 Fractional and negative indices .... 27 25, 26 Additional rules of differentiation .... 28 27 Differentiation of fractional and negative powers . 29 28, 29 Differentiation of quotients and rational functions 30 30 Illustrative examples .... Examples 31 The second differential coefficient . 32, 33 Mechanical and geometrical illustrations Examples 31 32 33 34 35 CHAPTER III SOME APPLICATIONS OF DIFFERENTIATION 34 Equation to a curve ...... 37 35 Parabola drawn through three points and having the direction of its axis given .... 38 36 Use of equation to a curve ..... 39 37 Equation to a straight line 39 38, 39 Condition of perpendicularity of two straight lines 40 40, 41 Tangents and normals to curves .... 41 Examples 42 42 — 47 Approximations 43 Examples 46 48, 49 Examples of maxima and minima .... 46 50 — 52 Theory of maxima and minima .... 48 53 Normal to a curve as a line of maximum or mini- mum length 50 Examples 51 54 — 57 The theorem of intermediate value ... 52 Examples 53 CHAPTER IV INTEGRATION 58 Mensuration of the triangle and circle ... 55 59 Examples of unknown functions with known dif- ferential coefficients 57 CONTENTS XI ART. PAGE 60 — 62 Meaning of "integration" 61 63 Use of the rules of differentiation ; . . 63 64 Integration of powers 64 Examples 64 65 — 68 Integration by change of the variable ... 64 Examples 67 CHAPTER V SOME APPLICATIONS OF INTEGRATION 69 — 71 Area under a curve . . ... 68 72 Definite integrals ... ... 70 Examples .71 73, 74 Parabola through three points . . . . 71 75 Area of a plane figure 73 Examples 74 76 — 80 Volumes of solids 75 Examples 79 81 — 83 Uniformly accelerated motion . 79 Examples 83 CHAPTER VI LOGARITHMS AND THE EXPONENTIAL FUNCTION 84 The logarithm as a function 84 85 Formulae connected with logarithms ... 86 86, 87 Differentiation of logarithms 86 88 Integration by logarithms ... 89 89—91 The exponential function 89 92 Evaluation of a certain limit ... 91 Examples 92 93 — 96 Applications of the exponential function to Chemistry, Electricity, and Mechanics . . 93 Examples 98 XU CONTENTS CHAPTER VII TRIGONOMETRIC FUNCTIONS ABT. PAGE 97 — 108 Definitions and simple properties of the functions 99 Examples 109 109 — 115 Differentiation and integration of the functions . 109 Exaiaples 114 116 — 118 Oscillatory motions 115 119 Creeping motion under large resistance . . 120 120 Alternating currents 121 Examples 122 CHAPTER VIII METHODS OF INTEGRATION 121—124 Partial fractions 125 Examples 129 125 — 127 A logarithmic formula 129 Examples 131 128 — 130 Integration by parts 132 Examples 135 131 — 133 Inverse trigonometric functions .... 135 Examples 139 134, 135 Change of variables by introducing trigonometric functions 139 Examples 143 Additional Exercises 144 CHAPTER IX VARIOUS RESULTS CONNECTED WITH ARCS OF CURVES 136 — 139 Formula for the length of an arc . . . . 146 140 — 146 Curvature 149 147 — 149 Area of a surface of revolution .... 152 Examples 15g CONTENTS XIU CHAPTER X THE DEFINITE INTEGRAL AS THE LIMIT OF A SUM AST. FAQE 150, 151 Approximate calculation of integrals . . . 157 152, 153 Approximate calculation of ir and e . . . 159 154, 155 Meaning of a definite integral . . . . 160 156 Applications to mensuration 163 CHAPTER XI SOME APPLICATIONS OF DEFINITE INTEGRALS TO MECHANICS 157 — 163 Centres of gravity and centroids . . . . 165 Examples 170 164, 165 Resultant thrust and centre of pressure . . 171 Examples 174 166 — 170 Moments of inertia and radii of gyration . . 174 Table of standard forms of integrals . . 178 APPENDIX I The graph of a rational integral function of the first degree 179 II Limits 1.80 III Indices and logarithms 188 IV The exponential limit 191 V The mensuration of the circle and the radian measure of angles 198 VI Trigonometric limits 202 VII Mechanical units 204 Index 205 CORRIGENDA Page 15, line 14, for «i, (0-1)" read '4 (O'l)." „ line 15, /or "0-95561" read "0-95661." „ line 18, for "Ex. 15" read "Ex. 16." „ 93, ftn., for "1894" read "1904." CHAPTER I INTRODUCTORY 1. Many quantities that we know how to measure are variable, for example, the temperature of the air, or the speed of a train. When a quantity can vary, it usually depends upon other quantities which can also vary. We shall consider some ex- amples of relations between the measures of variable quantities. (a) Weighing with a spring balance. Before any weight is hung on, the spring has a certain length. Let this length be b inches. Then 6 is a certain number ; it may be not a whole number, but that does not matter. When a weight is hung on,, the spring is stretched. When a weight of 1 lb. is hung on, let. the spring be stretched m inches, so that its length becomes. b + m inches. The number m would not generally be a whole number. When a weight of x lbs is hung on, let the length of the spring become y inches. If the weight is not too great the amount by which the spring is stretched is proportional to the weight, or we have. y — b:m = x:l, and this is the same as y = mx + b. The numbers x and y need not be whole numbers. The numbers m and b do not depend upon x or y, but the number y depends upon the number x. L. c. 1 2 CALCULUS [CH. I (b) Comparison of thermometric scales. Let C degrees on the Centigrade scale specify the same temperature as F degrees on the Fahrenheit scale. The diagram gig I lOQ (Fig. 1) gives at once the equation F-32 C Fig. 1. 180 100 F- 9 or F=^C + 32. 5 9 If for C, F, -, 32 we write as, y, m, b, the equation becomes 2/ = mx + b. Both X and y can be negative numbers. In these two examples we have used the same 32- lettei's X, y, m, b in order to bring out the similarity of form of the relations between the variable num- bers that occur. The letters always stand for numbers, but the quantities of which these numbers are the measures are different in the different examples. 2. Values assigned to two variable numbers, such as the X and y of the previous examples, can be shown graphically. We draw on paper two straight lines, which cut each other at right angles, one running from left to right, called the " axis of X," the other running up and down, called the " axis of y." The point where they meet is called the " origin." We choose a unit of length, a foot for instance, or a centimetre, or the dis- tance between two of the ruled lines if we are using squared paper. If a: is a positive number, we can find any number of points on the plane of the paper which are to the right of the axis of y, and at a distance x units of length from it. All these points lie on a straight line parallel to the axis of y. (One such line is dotted in Fig. 2.) In like manner, if y is a, positive num- ber, we can find any number of points on the plane of the paper which are above the axis of x, and at a distance of y units of length from it. All these points lie on a straight line parallel to 1.2] INTRODUCTORY the axis of x. The two lines meet in a point, which is distant X units of length to the right of the axis of y, and y units of length above the axis of x. The numbers x and y are called the ^ axis ofx Fig. 2. " coordinates " of the point. When x and y are chosen the point is fixed. We may say that the point " represents " the pair of numbers x and y. When one, or both, of the numbers x, y is negative, we can still represent the pair by a point. If a; is a negative number, — a; is a positive number, and we take the point, which represents the pair of numbers a; and y, to be at a distance — x units of length to the left of the axis of y. If j/ is a negative number we take the point to be at a distance — y units of length below the axis of X. In this way we find one point, and only one, which represents a pair of numbers x, y. The point is often called the point {x, y). 1—2 CALCULUS [CH. I 3. For example, in Fig. 3 the four points (2, 3), (2,-3), (-2, 8), ( - 2, - 3) are marked with crosses, the distance between consecutive ruled lines being taken as the unit of length. ^ """ ""~ ! 1 1 1 i '>\ \ (J. >•') c ^ X 1 ■2- tv r , ' ) V }] ' Fig. 8. 4. The axes of x and y divide the plane of the paper into four compartments or " quadrants." If from any point P we let y O N Fig. 4. 3-5] INTRODUCTORY 5 fall a perpendicular PN upon the axis of x, the straight line PN is called the "ordinate" of P and the straight line ON the "abscissa" of P (Fig. 4). The coordinates' x, y, if we disregard their signs, are the measures of the lengths-of the abscissa and ordinate. The coordinates, with their proper signs, tell us not only how long to make the abscissa and ordinate, but also in which of the four quadrants the point P lies. 5. We go back to the equation y = lx+32, in which x and y are the two numbers which specify the same temperature on the Centigrade and Fahrenheit scales. We mark 1 1 1 1 1 1 1 1 1 r 1 1 1 1 1 1 1 1^ r 7 t 1 T 1 -^i. J 4- z r^ 1 t t 1 %r I t 7 t _i T 1 ■J i T 1 lu 10 10 Fig. 5. on squared paper some of the points whose coordinates satisfy CALCULUS [CH. I the equation. For instance some corresponding values of x and y are given by the table -10 14 -5 23 32 5 41 10 50 15 59 We find that all the points which have these coordinates lie on a straight line. Part of this line is shown in Fig. 5. It is easy to prove formally that every pair of values of x and y by which the equation can be satisfied is represented by a point on this line, and that every point on this line represents a pair of numbers X, y which satisfy the equation. But we shall omit the proof for the present. The line is said to be the graph of the expres- 9 sion -^33 + 32 to which y is equal, or shortly " the graph of 9 2/ = V x + 32." It is easy to prove formally that, if m and h are independent of x, the graph oi y = mx + 6 is a straight line*. 6. If a body, such as a stone, moves over a distance s feet 8 in t seconds, its average velocity during this interval is - feet t per second. The body may move in such a way that this is the same for all values of t. It then moves " uniformly." If we Fig. 6. * A formal proof will be found in Appendix I. 5-7] INTRODUCTORY 7 write V for the constant value of — , the velocity of the body is t V feet per second. If we write y for s and x for t, we have y = vx. The graph of this equation is the " distance-time " graph for the body. Let A and B be two points of the graph, £c^, y^ the coordinates of A, XjB, t/jg those of B. We take Xjg to be greater than x^ as in rig. 6. The number Xjg — x^ is the measure in seconds of a certain interval of time. The number y^ — y^ is the measure in feet of the distance over which the body moves during that interval. The fraction Vb-Va Xb-Xa is the measure in feet per second of the average velocity in this interval. If, as above, y = vx, where v is independent of x, y^ = vx^ and y^ = vx^ , so that the fraction is equal to v, and therefore has the same value for every interval. 7. If we have any straight line graph we can choose two points A and B on it, and form the fraction — — —, and this Xs- x^ fraction has a simple graphic interpretation whether the straight line is a distance-time graph of a moving body or not. We take B to be further to the right than A, so that Xb>Xa. If B is higher up than A, y^ > y^ (Fig. 7 a). If A is higher up than B, yjB < y^ (Fig. 7 /8). In the first case the graph goes up to the right, and the fraction — — — is positive. In the second case it goes down to <«b-Xa 8 CALCULUS [CH. I the right, and the fraction is negative. If we disregard the sign the fraction gives us a measure of the steepness of the graph. It is as if the straight line ran through the edges of a set of steps so that A and B are edges of two consecutive steps (Fig. 8). Then Xjg — x^ is the measure of the breadth of a step, and the absolute value of y^ — yA (sign disregarded) is the measure of the height of the step. The absolute value of the fraction — — — (sign dis- regarded) tells us how steep the line is, and the sign of the fraction tells us whether the line goes up to the right or down to the right. The fraction with its proper sign is called the "gradient" of the line. Fig. 8. Since ^"^~^^ = ^-""^^ , the gradient is the same whether B Xj^ X^ X^ — Xj^ is to the right of A or to the left of A. 8. Let the straight line be the graph oi y = mx + h. We have yji = mxs + h, yA = 'mx^ + i, ■so that yB-yA = in {^B - Xa)> ^nd y^-y^ =m. x^ — x^ Hence the number m in the equation y = mx + 6 is always the gradient of the corresponding graph. The graph goes up to the right or down to the right according as m is positive or negative. If the straight line is the distance-time graph for a body moving uniformly, the gradient is the measure, in appropiiate units, of the velocity of the body. 7-9] INTRODUCTORY 9 If the straight line is parallel to the axis of x, y is the same at all points of it. Then ya — VA — 0, and the gradient is 0. Examples Draw the graphs of the following, (1) — (17), and determine the gradient in each case : — (1) y = x, (5) y=-x + l, (9) 2/ = 2a:, (2) y=x+l, (6) y=-x-l, (lO) y = 2x + i, (3) y = x~l, (7) y = x + 2, (11) y = 2x-l, (15) y=-^x-l, (4) y=-x, (8) y=-x-S, (12) y=-2x, (18) y=-l«>, (18) y=-'2x-S, (14) y=-x, (17) y=-^x + 2. 9. When a stone is let fall it begins to move very slowly, but after it has been moving for a little time it moves more quickly. The number of feet through which it falls in a few seconds is not always the same multiple of the number of seconds. In an interval of t seconds, reckoned from the instant when the stone is let fall, it moves a certain distance, which we take to be s feet. Neither t nor s need be a whole number. Now it is found that, apart from a correction depending on the resistance of the air, s = {U-l)f. Put y for 16-1 and X for t. We get y = x'. This equation is quite different from any that we have had before. We proceed to draw tlie corresponding graph. As before, we make a table, thus ; — • 0-1 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9 0,01 0-04 0-09 0-16 0-25 0-36 0-49 0-64 0-81 for convenience we take the distance between two consecutive ruled lines on the squared paper to be one-tenth of the unit of length. We mark the points which have the above coordinates and draw a smooth curve through them. This curve is part of 10 CALCULUS [CH. I the graph of y = a?. To complete the graph we should have to find the values of y that correspond to negative values of x, and y 1 V4- T 7 \ ' V 2 \ 1 S I \ V I^ ^ =ffi=^-.:..^f=====i -1 O 1 Kg. 9. to values of x that are greater than 1. In Pig. 9 the graph is drawn for values of x that lie between — 1 and 1. In this example the graph is a curve called a "parabola." Examples 1. Plot the graph oi y = x^ between x = l and x = 2. 2. Plot the graph oi y= -x^ between x= -1 and x = l. 3. Make a table of the values of :^ x^ when x has the values -5, -4, - 3, - 2, - 1, 0, 1, 2, 3, 4, 5. Mark on squared paper the corresponding points of the graph of j;=— x' and draw the graph. 10. In the case of the falling stone we had s = (16-l)i!^ y= 16-1 x = t. y= If we take 16'1 feet as the unit of length, the stone falls y units of length in x seconds, and the number y is the square of the number x. Let A, B be two points on the graph of y = tji?. In a;^ seconds the stone falls y^ units of length, in x^ seconds it falls y^ units of length. We take x^ to be greater than x^. Consider the interval of time which begins at the instant x^ For shortness write h for x^ — x^, so that Xg = Xji^ + h. Then 9-11] INTRODUCTORY 11 seconds after the instant when the stone was let fall, and ends at the instant x^ seconds after the instant when the stone was let fall. During this interval of Xg — x^ seconds the stone falls Vb ~ Va units of length, and its average velocity is — — — units x^i x^ of length per second. shortness wr Va = ^A^ and y^j = «^b = (a^^ + Kf = a;/ + Ihx^ + W, and VB-VAj^I^'^h^A^^'-^A'Jhx^ + h-^ j^_ Xg—x^ h h During the interval of h seconds, beginning at the instant specified by x^^, the average velocity of the stone is 2a;^ + h units of length per second. This average velocity depends not only on the value of x^ but also on the value of h. If we keep x^ always the same, and take a smaller value for h, then a still smaller value, and so on, we see that we can bring the expression ix^ + h as near to 2a;^ as we please. We may express this by saying that, as h tends to zero, Ix^ + h tends to 2a;^ as a limit. Keeping x^ always the same is keeping the initial instant of an interval always the same. Diminishing h is shortening the interval. We have learnt that, as the interval is shortened, the initial instant remaining the same, the average velocity tends to a limiting velocity. When we speak of the velocity of the stone at the instant in question we mean this limiting velocity. Further we have learnt that the velocity at the instant specified by x^ is Ix^ units of length per second. Our unit of length was 16-1 feet, and therefore the velocity is (32'2)a3^ feet per second. In other words, the velocity of the stone at the instant which is t seconds later than the instant at which it was let fall is (32-2) t feet per second. 11. We consider the matter more generally by thinking of the graph of y = x^ without regarding it as a distance-time graph. As before let the ai-coordinates of two points A and B be a:^ and x^ + h. Formerly we took B to the right of A, so that h was 12 CALCULUS [CH. I positive, and we found that the gradient of the straight line AB was 2a;^ + h. But this result is unaltered if B is to the left of A, so that h is negative, for the algebraic work by which we found the result remains the same. We found that, if h is positive, 2a;^ + h tends to 2a;^ as a limit when h tends to zero. If h is negative 2a3^ + A is less than 2ce^. If we take a value of h, negative but nearer to zero than before, then a,nother nearer still, and so on, we see that we can bring 2a;^ + h as nearly up to 2a;^ as we please. Thus 2a;^ + h tends to the same limit 2a;^ whether h is positive or negative. Bringing h nearer and nearer to zero is bringing B nearer and nearer to A. The gradient of the secant, or cutting line, AB is 2x^ + h. It can be greater or less than 2a!^, but cannot be equal to 2x^ for any secant drawn through A. As B is brought nearer and nearer to A, either on the right-hand side or on the left, the gradient tends to '2x^ as a limit. A straight line drawn through A, and having the gradient 2a;^, would not cut the curve again, but if we in- creased or diminished the gradient ever so little it would cut the curve. The straight line in question touches the curve at A. It is the " tangent " to the curve at A (Pig. 10). The number 2a3^ is the gradient of the tangent to the curve at A. "We may call it the " gradient of the curve " at A. We have learnt that the gradient of the parabola, given by y = x^, at the point whose coordinates are x and a?, is 2a;, 12. The process here described is of very general application. It may be summed up in the rule : — Let A, B be two points of a curve. Find the gradient of the straight line AB. The limit to which this gradient tends, as B is brought nearer and nearer to A, is the gradient of the curve at A. . Fig. 10. 11, 12] INTRODUCTORY 13 The business of the Difierential Calculus is to determine in each case the limit to which the gradient of the secant tends. When we determine such a limit we differentiate. In the case of the parabola we diflFerentiated ay'. The result was 2a!. In the case of a straight line graph we do not need to bring B nearer to A in order to determine the gradient. The gradient is for the graph of y = a, it is m for the graph of y = mx + b. We may say that, when we differentiate a, the result is 0, and when we differentiate mx + h, the result is m, the numbers a, b, m being independent of x. Examples X. A moving body passes over s feet in t seconds, and s and t are con- nected by the equation s = t^. What is the velocity of the body at the end of the fifth second ? What is the average velocity of the body during the fifth second ? [Results : 10 ft. per sec, 9 ft. per sec] 2. As in Ex. 1, a body moves according to the law s = t^. Find its average velocity in the first tenth, first hundredth, first thousandth of a second, beginning at the end of the fifth second. 3. The point (1, 1) on the parabola, given by the equation y=x^, is joined to the point {1 + h, (1 + ft)^}. Find the gradient of the secant when ft has the values 0-1, 0-01, 0-001, and -04, -0-01, -0-001. 14 CALCULUS [CH. I Additional Exercises Some additional examples of graphs and computing are placed here. These should not all be worked out before reading the rest of the book. The student will find that it is better to do one or two of these exercises every day. 1. Plot the graph of j/=logioa; from a; = 0'l to 1 = 10 by giving to x the values 0-1, 0-2, ... 1 and 1, 2, ... 10. a. Plot the graph of ^ = 10"= from x= -1 to x=l by giving to x the values -1, -0-9, -0-8, ... -0-1, 0, 0-1, 0-2, ... 0-9, 1. 3. Make a table of the values of x logio ( 1 + - ) by giving to x the values 1, 2, 3, ... 9. Proceed to make a table of the values of (l + - ) for the same values of x, and plot the graph of y = (l-\ — j between x = l and x=9. 4. Do the same as in Ex. 3 with the values 10, 20, 30, ... 100 for x. 5. Plot the graph of y=- by giving to x the values O'l, 0-2,'... 0-9, 1, 2, and the same values with minus signs ( - 0-1, ...). e. Draw on one piece of squared paper the graphs of y = —x^ and y = T^(7x + 2). Find the ^-coordinates of the points where they meet. [Besults : 2-8, -0-4, -2-4 approximately.] Note that this method gives the approximate solution of a cubic equation xS-7x-2 = 0. 7. Solve approximately the cubic equations s^-9x + 7=0 and 3a;3-5x + l=0. 8. Plot the graph of y=x + - from a;=0-5 to a;=l-5 by giving to x the values 0-5, 0-6, 0-7, ... 1-5. 9. Plot the graph o{y=x(l- 2x)^ from a; =0 to x= 1 by giving to x the values i 1 1 1 A 1 7 2 8 5 11 values ^g, g, ^, g, ^g, -, -, -, _, _, _,i. 10. Plot the graph of 2^=x(lB-2x) (15 -2a;) from x=0 to x = 10 by giving to X the values 0-5, 1, 1-5, 2, ... 9-5, 10. 12] INTRODUCTORY 15 11. Plot the graph of y = x {13-2x) (15-2x) from x = 6-5 to a: = 7-5 more minutely than in Ex. 10, by giving to x the values 6-5, 6'6, 6'7, 6-8, ... 7-4, 7-5. 12. Given logioe = 0-4343, make a table of the values of c"^ for x=0, O'l, 0-2, 0-3, ... 0-9, 1. 13. Do the same for e~*. 14. Plot a graph oiy^el' from a;= -1 to a; = l. 15. Plot a graph oty=X- e^" by taking x = 0, 0-5, 1, 1-5, 2, 2-5, 3, 8-5, 4, 4-5, 5. 16. Given y=^{l-x^}, calculate, to six places of decimals, the values of 2/ which correspond to the values -O'S, -0-4, -0-3, -0-2, -01, 0, 0-1, 0-2, 0'3, 0-4, 0-5 of X. Denoting these values by yi, y2, ... J/ii, find the value to five places of decimals of 3. (O'l) {3/1+2/11 + 2 (!/3 + y6 + y7 + 2/9)+4(2/2+V4 + 2'6 + J/8+2/io)}- [Result: 0-95561.] 17. Given y = - , calculate, to six places of decimals, the values of y which correspond to the values 1, 11, 1-2, ... 1-9, 2 of x. Proceed as in Ex. 15. [Result: 0-69315.] We shall use the Results of Bxs. 16 and 17 in Ch. X. CHAPTEE II DIFFEEENTIATION 13. We have considered a few examples in which one variable number y is expressed in terms of another variable number x by an equation. These numbers may be thought of as the measures of certain variable quantities in terms of appropriate units, but they need not. The expression by means of which the value of y can be written down when a value of x is chosen may be much more complicated than any of those which we considered. But, whether the expression is simple or complicated, we think of it as equal to another number y, which is variable when x is variable. When we think of an expression containing x as being itself a variable number, which takes various values according to the value given to x, we call it a " function " of x. For instance a?, logio X, sin x are functions of x. We use the notations f(x), F (a;), ^ (x) &c. to denote functions of x. It is important not to think of the symbol f in /'(a;) as a number, but to regard the expression "/(a:) " as an abbreviation for " a function of a." When we draw, or plot, a graph by using an equation of the form y =f{x), we draw, or plot, the " graph of the function.'' All the functions that we shall have to consider possess graphs*. 14. We found that we could determine the gradient of the graph at a point by a certain process whenever we could carry * For a limitation implied in this statement see Appendix II. 13-15] DIFFERENTIATION 17 out the process. The process was this : — Let a;^ be the x of the point. In the expression f(x), to which y is equal, substitute as^ for X. In the same expression substitute x^ + h for x. Form the difference / {x^ + h) ~f (x^), and divide it by h. Determine the limit to which the quotient /{x^ + h) -/(xa) h tends as h is diminished towards zero, if it is positive, or as h is increased towards zero, if it is negative. In the result drop the suffix A. Then we have the gradient of the graph at any point (x, y) on it. We may omit the suffix A throughout the process, but when we do this we must keep it in mind that throughout the process a; is a constant number and h is the only variable. 15. We shall now write Asc instead of h. The symbol A (Delta) is not to be thought of as a number multiplying x, but the expression Ao: is to be thought of as itself denoting a number. It means "a number added to x." This number Aa; may be positive or it may be negative. When we replace x \D.f{x) by x + h or x+ Aa;, obtaining the expression /(a; + Ax), we may use a number Ay to stand for the difference /(a; + Aa;) —f(x), so that y+^y=f{x+^x), y being the number to which_/(a;) is equal ; and then Ay is " the number that is added to y when Aa; is added to x." The quotient f{x + Ax) -f{x) is the same as A.x Ay Aa;' and we have to determine the limit to which this quotient tends when Aa: tends to zero. We write dy dx 18 CALCULUS [CH. II for this limit. This symbol is not to be thought of as a fraction or quotient, but as the limit to which the quotient — tends as Ace tends to zero. We need not attempt to give a meaning to dy or to dx. The limit expressed by the symbol -^ is called the " differ- ential coefficient of y with respect to x." 16. When the limit is found it appears that it depends upon the value assigned to x. After finding it we may again regard x as a variable. For instance we found that when y = 9?, -— = 1x, dx and 1x is a function of x. When we wish to think of the differential coefficient of f{x) as a function of x we denote it by f'(x). The function J '{x) is often called the " derived function " of/(a;). 17. The results which we have found so far may be written in the notation of differential coefficients as follows : — (i) If a is independent of jc, ;j-=0, (ii) If m and b are independent of x, — i—j : = m, In the result (ii) is included the result j-=l, which is obtained by putting m=l, 6 = 0. This result has been proved byfindingthe gradient of the graph of y=x. It may not be iuf erred from the form of the symbol — by thinking of this symbol as a fraction of which the denominator is equal to the numer- ator, because the symbol is not a fraction with dx for numerator and dx for denominator. 18. We know that one use of differential coefficients is to find the gradient of a graph. We now illustrate some further uses of them. (a) Falling body. We consider a stone let fall. We know that, apart from a correction depending upon the resistance of the 15-18] DIFFERENTIATION 19 air, the distance s feet through which it falls in t seconds is given by the equation In t +At seconds it falls through s + As feet, and we find As=(16-1)(2«+A«) A<. During the particular interval denoted by A« seconds it falls As through As feet, and its average velocity in this interval is — feet As per second. As A< tends to zero the number -— tends to a limit A« which is -r . We find at I = (32-2). The velocity (32-2) t feet per second is the velocity of the stone at the instant specified by t, that is to say t seconds after it was let fall. (6) Speed of moving body. More generally, if a body is in motion it moves a distance s feet in t seconds, and s + As feet in As < + A« seconds. The number — - is the measure in feet per second A« '^ of its average velocity in the interval denoted by At seconds, and ds the limit -^, to which this number tends as At tends to zero, is dt' the measure in feet per second of its velocity at the instant specified by t. (c) Bate of change in general. Still more generally, if 5 is a number, which is the measure of a variable quantity in terms of some unit, and the instant at which the quantity is measured by g is < seconds later than some chosen instant, g- is a function of t, and the differential coefficient ~ measures the rate per second at which the quantity measured by q is increasing. For example, in a vessel containing some water and some ice, j may be the number of lbs. of ice at the instant speciflced by t. If ~ is positive, 2—2 20 CALCULUS [CH. II it measures in lbs. per second the rate at which water is being converted into ice, or the rate of freezing. If -^ is negative, --^ measures in lbs. At at per second the rate at which ice is being converted into water, or the rate of thawing. dx (d) Tangent to a curve. We have already seen that is the gradient of the tangent to a graph at the point (x, y). Let the tangent at a point P above the axis of a; meet this axis in T, let PN be the ordinate of P, and sup- pose that the curve goes up to the right (Fig. 11). The fraction number of units of length in thelength of N P number of units of length in the length of N T is the gradient of the tangent at P. A similar result holds in any case if we give the right sign to the fraction. (See Ch. IX. ) It appears that when the gradient is given we can draw the tangent. For example, take the parabola given by the equation y = x^. We have -p = 2a;. Now PN contains x^ units of length when ON contains s units 18, 19] DIFFERENTIATION 21 of length. Let TN contain z units of length. Then — = 2x, or i:=^x, and therefore TN is half of ON. To construct the tangent at P, draw the ordinate PN, bisect the abscissa ON in T, and join TP. The straight line TP is the tangent to the parabola at the point P (Fig. 12). Examples 1. A body is moving in such a way that the distance s feet passed over in any time t seconds from the start is proportional to the square of the velocity at the instant specified by t. Express this fact by an equation con- necting — with s. a. A body is moving in such a way that its velocity, v feet per second at the instant specified by t, is increasing at a uniform rate. Express this fact by an equation containing -y- . 3. A body is moving in such a way that its velocity, v feet per second at the instant specified by t, is diminishing at a, rate proportional to v. Express this fact by an equation. 4. A tank is being emptied in such a way that the rate at which the water flows out is proportional at any instant to the amount of water left in the tank at that instant. Express this fact by an equation. 5. One substance is being transformed into another according to the law that the rate of transformation per second is proportional at any instant to the amount that has not been transformed at that instant. Express this law by an equation. 19. We shall be able to proceed more quickly afterwards, and we shall be more certain that our work is correct, if we take a little time to think exactly what it is that we mean when we say that a function of h tends to a limit as h tends to zero. When in § 11 it is said that, as A (supposed positive) is diminished towards zero, 2x^ + h tends to 2x^ as a limit, it is meant that, without actually making h = 0, we can make 2x^ + A as near to 2x^ as we please. Thus if we want to make 2x^ + h differ from 2xj^ by less than Y^ivxm ^^^ ^^ have to do is to make h less 22 CALCULUS [CH. II than lOOoo TTTT- There is never any vagueness about a limit. The limit in this case is 2x^, not 2x^ + a very small fraction, or 2x^ — a very small fraction. If anyone thought it was 2x^ + a very small fraction _/^ we should only have to take h less than y to prove that 2x^ + h can be brought nearer to 2x^ than 2a;^ +f. Without taking h equal to we can bring 2a;^ + h a,s near to 2x^ as we please, but we cannot mak« 2x^ + h equal to 2x^ . In a case like this the limit is precisely known, but it is not a value of the function for any value which the variable can have. In fact 2x^ + A is a function of h, and h may have any value except 0, it cannot have the value because we have divided by it. So 2xJ^ + h cannot have the value 2x^ , but, as said before, it can be made to dififer from this value by less than any number however small. On the other hand there are cases in which the limit is a value of the function. In the example which we considered the function was 2x^ + h and the limit was 2xj^. The difference (function) - (limit) may be positive or it may be negative. The property which dis- tinguishes the limit from all other numbers is this : — The absolute value of the above difference (sign (^sregarded) can be made as small as may be wished by bringing the variable h near enough to zero. 20. As another example we consider the differentiation of a?. We have (x + ft)3 = x3 + 3Ax2 + 3/j2x + fts, so that (x + hf -x3= Shx^ +3h^x+ h\ and (£±^!z£'=3a;2 + 3&t + ft2 n and we can show that the Umit is Sx^. As we shall use a different method presently, it will be sufficient here to take the case where x and ft are both positive. We have to show that, by taking ft small enough, we can make Zhx+h? as small as we please. We may begin by taking ft to be smaller than X. Then }fi. dx ax The result may also be written in the form -^=n- when y-x^. 23. The formula for differentiating x^ (n a positive integer), combined with the rules (i) and (ii) of § 21, enables us to differentiate a large class of functions. A function of this class is expressed by a sum of terms, each term is the product of some constant number and a positive integral power of x. The general form of such a function is oa;" + Saj""' + cx'^-^ + ... +px + q, where a, b, c, ... p, q are numbers independent of x, and w is a positive integer. Such a function is called a " rational integral function of the nth degree." The function mx + b which we considered in the last Chapter is a rational integral function of the first degree; it is often called a "linear" function, because its graph is a straight line. If any term of a rational integral function has a minus sign prefixed to it, the differential coeflBcient has a minus sign prefixed to it. For example, if in the above expression ax^ were — as" the differential coefficient of that term would be - Ma!""^. The co- efficient a is — 1. 22-24] DIFFERENTIATION 27 Examples 1. Differentiate the following (1) — (14) : — (1) -a:2, (2) ia;2, (3) x^+ix, (4) a:2-2x, (5) ir<:2 + 3a;, (6) -Ix^ + Zx, (7) x(l-x), (8) x2(l + 2x), (9) x^-Zx, (10) ^i?, (11) -x3, (12) 2a;3-3a!2, (13) a;3 - 4a;2 + lOx, (14) |a^-|a; + i. 2. The tangent at a point P to a curve, given by ky=x^, where k ia independent of x, meets the axis of x in T, and the ordinate of P meets the axis of X in N. Prove that OT is half of ON. 3. The curve as given by ky=x^ and the notation is the same as in Ex. 2. Prove that OT is two-thirds of ON. 24. We shall show presently that the formula — ^ — ■' = ma;""^ holds in the cases where n is fractional and where n is negative. We call to mind the meanings of fractional and negative indices. We know that the meaning of an index in general is fixed by the equation dU Jj — iX^ ■ called the " index law." We know that, if ji is a positive integer, this law shows that a;" means the product of n factors, each equal to x. We know also that if m. is a positive fraction of the form -, where p and q are positive integers, this law shows that a;", or a:', is the /)th power of the 5th root of a;, and that it is also the gth' root of x^. If p and q have any common divisor other than 1 it may be removed, so that the fraction - may be taken to be in its lowest terms, and when this is done, it is t understood that, if q is even, x is positive. If p is even a;« is always positive. If p and q are both odd a:« has the same 28 CALCULUS [CH. II sign as x. Further the index law shows that, if n is negative, and we put n = — m, so that m is positive, a;" or x~™ means — . 25. We next introduce two additional Rules of diflferentia- tion. We number them consecutively with the Rules in § 21. (iv) If we are given y a,s a, function of z, and « as a function of X, we may regard y a,s a, function of x. Then we have the Rule dy _ dy dz dx dz dx' It is important to notice that we may not infer this by cancelling the dz's, as if the expressions were fractions. In order to prove the Rule we must remember what the various expressions mean. When X is changed to a; + Aas, z is changed to a + Aa, and then y is changed to y + ^y. We are supposed to know that --^ and As As: . dy -— tend to limits when Ax tends to zero. These limits are -^ and Ax dz -r;- . Hence the product — ^ - — tends to a limit which is -^ -;-. dx A« Ax dz dx But in the product v^ -r- we may cancel the Az's because the two factors are quotients, and therefore this product is equal to 7e knc dy dz —2- . We know therefore that — ^ tends to a limit, and that this A.X Ax limit is , , . dz dx This Rule is known as the Rule for differentiating a " function of a function." (v) If 2/ is a function of x, x is also a function of y. Some- times it is easy to find -= . Then we can write down the value of^bytheRule^^=l. dx dx ay 24-27] DIFFERENTIATION 29 It is important to notice that the Rule may not be inferred by cancelling the da^s and dy's. When X is changed to a; + Aa;, y is changed to y + A_iy. Hence not only is Ay the number that is added to y when Aa; is added to X, but also Aa; is the number that is added to x when Ay is Aw Aa; added to y. Therefore in the product r^ t— 'we may cancel the Aa;'s and the Ay's, and the value of the product is 1. Now we Aa; are supposed to know that as Ay tends to zero — tends to a fit* ^^"iJ 1 limit which is -^ . Hence — ^ tends to a limit which is -7- . ay Aa; dx dy 26. The variable with respect to which we differentiate is usually called the "independent variable," and a variable which is regarded as a function of the independent variable is usually called a " dependent variable." Tbe Rule (iv) is tbe rule for changing the independent variable. The Rule (v) is the rule for interchanging the dependent and independent variables. 27. We can now verify the general formula d\vf-). — V^ = wa;"-^ . dx First let n = -, where p and q are positive integers. P Put y = a;" = xi, then x^ = y*, and we may put z = x^, z = y^, 1 ,, dz „ ^ dz and then -=- = px^~^ , -z- = qy^ ^. Now dy dy dz 1 dz px''~^ p x^ y dx dz dx dz ' dx qy^~^ q' y^' x dy p qx q The formula is now verified for all positive integers and positive fractions. 30 CALCULUS [CH. II Next let n = — m, where m is a positive number, integral or fractional. Put y=x"' = 03-" = — - . Then »"«=!. and -\ — '-=ma?"-\ ax Now we use the rule for differentiating a product, and find dla^y) „ , dy dx " dx But, since a;" y = 1 , ^ ^ ' = 0. Hence x™' -^ + mx^-^ y = 0, or ^ = _m-=- wia;-™-' = m!''-^ dx X The formula is now vei-ified for all integral and fractional values of n, positive or negative. In applying the formula and the Bule (1) of § 21 to differentiate ax" we must pay attention to the signs. For example, if we had to differentiate - x~', the result would he ^x~i . The special case in which n= - 1 is very important. It gives the result 1©.-!. ix a;2 28. We consider some additional formulae of a general character, which, however, are hardly distinct enough from the previous Rules to be regarded as new Rules. (i) Let y be given in the form - , where m is a function of X which we know how to differentiate. We have dy^d^dM ^^^ %^_Jl dx du dx du v? ' du 1 d/u, hence -/- = ; -=- . dx w dx 27-30] DIFFERENTIATION 31 V (ii) If y is given in the form — , where v and u are functions of X, we may regard y as the product of v and — , and apply the rule for differentiating a product. We have d- dy \ dv u 1 dv 1 du dx u dx dx u dx v? dx dv du dx dx " 1? This result is often called the rule for " differentiating a quotient.'' 29. Among the functions which can be expressed in the V form — are those in which both v and u are rational inteeral •w functions of x. Such functions are described as " rational fractional functions" of x. Rational fractional functions and rational integral functions together constitute the class of " rational functions.'' We know now how to differentiate every rational function. We know also how to differentiate many other functions, involving fractional powers. Such functions are not rational functions. It is necessary to practise differentiation without thinking about its applications, so as to be able to differentiate without making mistakes. 30. We work out some examples. 1 (i) LetM=l-a;2, and let y = u~i = Then dy dy du dx du dx = ( - ^ ""^) ( - 2a; ) =xu-^ X ~(1-X2)#' 32 CALCULUS [CH. II (ii) x+^{xi-l). Let ii=a!2-l, and let y=x + ui- Then dy^^^d^^^^dj^^ dx dx du dx = 1+ Qm~^) {2x) = l + xu-i :1 + - This result may also be written dy _x + ^{x^-l) _ y di~ v/(a;2-l) ~v/(a:2-l)' (iii) a; (1 - 2a;)2. Let ^ = a; (1 - 2a:)2. Apply the rule for differentiating a product. We get Put l-2.=u. thenliil^M!L=^ = ^*5 ax ax au dx = 2«(-2)=-4(l-2a;). Hence ^1 = (^ " 2*)^ - ^^ (^ " ^^^^ = (^ " ^a;) (1 - 6a!). (iv) Let v=x, u=x^ + l, y = X V u' dv , du VLiA/ U/tlr and therefore or <: 0. 6. Prove that the graph of y=x^ is convex to points that are above it when X is negative, concave when x is positive. CHAPTEK III SOME APPLICATIONS OF DIFFEEENTIATION 34. Before proceeding with the theory we consider some applications of the results so far obtained. We have already seen what is meant by a function, and how a function may be represented by a graph. Instead of thinking of a function, such as sc^, or an equation, such as y = a;^, and the corresponding graph, we may think about a curve, such as a circle or a parabola. The curve is the locus of a point which moves according to some law. For instance, a circle is the locus 38 CALCULUS [CH. Ill of a point which moves in such a way that its distance from a fixed point is always the same. To express this law, we take the fixed point as origin, and the distance of a point on the circle from the origin as r units of length, take any two straight lines drawn through the fixed point at right angles to each other as axes of X and y, and draw the ordinate PN of any point on the circle. Since the angle at N is always a right angle (Fig, 14), we have, in every position of P or {x, y), The parabola arises in connexion with the problem : — One side of a rectangle being given, it is required to find the other side so that the rectangle may be equal in area to a square. In Fig. 15 let AB be the given side, ACDE the square, ABMN the re- quired rectangle, and let MN, CD meet in P. If we keep A and B fixed, and let C move on the line AB, the locus of P is a parabola. If we take the axes of x and y along AB and AE, and the length of AB a& k units of length, we have the equation ky = a?. If k = \ we have y = as^ as in § 9. M B Fig. 15. When, as in these examples, we express the geometrical condition, by which the points of a curve are distinguished from other points, by an equation connecting the coordinates of a point, this equation is called the "equation to the curve." 35. When the equation to a parabola is given in the form the axis of y is the " axis " of the parabola, the origin is the " vertex " of the parabola, and the axis of x is the " tangent at the vertex." The curve lies above or below the tangent at the vertex accordibg as h is positive or negative. The equation 34-87] SOME APPLICATIONS OF DIFFERENTIATION 39 is the equation to a, parabola, of which the axis is the straight line whose equation is a;=Xo, and the vertex is the point {xq, y^. Writing this equation. we see that it has the form y = ax'i + px + y (1), where u,, ;8, y are independent of x. Conversely this equation can be written 1{>-HT)\'H^)'- and therefore every equation of the form (1) represents a parabola with its axis parallel to the axis of y. In general we can find a parabola having its axis parallel to the axis of y, and passing through three given points. Let {xi, yi), (x^, yi), (x^, y^) be the points. We have to find three numbers a, j3, y so that axi^ + pxi + y=yi, aX2^ + pX2 + y=y2, aX3^ + pxs + y=ys (2). From these equations we find y2-yi=a{x2^-xi^)+p{x^-xi), y3-yi='i{x3^-x^)+p(x3-x2), and therefore , ^^^^=a(x2 + a!i) + /3, ^^^^=a.{x3+X2)+P (3), whence ^B _ ^i^i = „ (^3_^J (4). Equation (4) gives a. When a is found from it, either of the equations (3) gives /3. When a and /i are known either of the equations (2) gives y. It is assumed that no two of the given x'b are equal. 36. The equation to a curve expresses a geometrical property which is common to all the points of the curve. By means of the equation and some analytical process, e.g. differentiation, we may discover other properties of the curve. For example in the case of the parabola whose equation is y = oi^ we found that the tangent bisects the abscissa (§ 18). 37. The equation to a straight line expresses the condition that the straight line has the same gradient at every point. The equation to the straight line which passes through the point (0, 6) and has the gradient m is y = mx + b. 40 CALCULUS [CH. Ill The equation to the straight line which passes through the point (a^, yi) and has the gradient m is The equation to the straight line which passes through the points (iBi, yi) and {x^, y^) is for the fraction — — — is the gradient of the straight line (§ 7). x^ ~ x^ 38. Two straight lines at right angles to each other being drawn through the origin, it is required to find an equation con- necting their gradients. The line Ox drawn to the right lies in one of the four angles formed by the two straight lines, and therefore one of them goes up to the right, and the other down to the right. Let the gradient of the one that goes up to the right be m, that of the other m'. Then m is a positive number, and m' is a negative number. Let P be a point on the line that goes up to the right, PNQ a straight line at right angles to the axis of x, and let the a; of P (and Q) be positive (Fig. 16). The angles OPN, NOP are together equal to a right angle, and so are the angles NOP, QON. Hence the angle OPN is equal to the angle QON. Also, in the triangles OPN, QON the angles at N are equal, being right angles. Therefore the triangles OPN, QON are similar, and QN : ON =ON : PN. Now the 2/ of Q is negative, and - y^ is the number of units of length in the length of NQ. We have therefore -yn Xp , 1 — ^^=-^, or -m=~. Fig. 16. 37-41] SOME APPLICATIONS OF DIFFERENTIATION 41 The required equation is nvm! = — 1. 39. The equation connecting the gradients is the same for any two straight lines which are at right angles to each other, whether they pass through the origin or not. If is the equation to a straight line passing through the point (2/-2/i)m + a;-x, = is the equation to a straight line passing through the same point and cutting the other at right angles. 40. We can form the equation to the tangent to a curve at a point. Let y =f{x) be the equation to the curve, x^ and/(£Ci) the coordinates of the point. The gradient of the tangent is the limit to which the quotient f{«h. + K) -/(a^i) h tends as h tends to zero. This limit is obtained by substituting a?! for X in the derived function /' (x). We may write /' {x^ for this limit. Then the equation to the tangent is y-yi=f {xi) (x-iBi). To find/' (asi), first differentiate /(sc), then substitute x^ for x. A straight line drawn through a point of a curve at right angles to the tangent at the point is called the "normal" at the point. If y =f{x) is the equation to the curve, the gradient of the normal is — j^-. — r , where /' (xj) is the gradient of the tangent, as has just been explained. The equation to the normal is therefore (y -2/i) /'(«',) + (a: -a'i) = 0. 41. If -;^ = at a point of a curve we know that the ax tangent to the curve at the point is parallel to the axis of x. If the point is on the axis of x, the tangent is the axis of x. For 42 CALCULUS [CH. Ill example, the tangent at the origin to the parabola whose equation is 2/ = ar* is the axis of x. li y= ijx we have in general ~ = „— p , but this formula does not determine the value of -y- at the origin because it directs us to divide by 0, which we cannot do. This difficulty is met by observing that in general -=- = ^Jx, and when a; = this equation ay becomes -5- = 0. The tangent at the origin to the curve whose equation is y= Jx is the axis of y. Whenever the evaluation of -^ would require division by 0, i.e. whenever -j- = Q at a point of a curve, the tangent to the curve at the point is parallel to the axis of y. Tf the point is on the axis of y the tangent is the axis of y. Examples 1. Prove that, if y = ax^ + ^x + y is the equation to a parabola passing through the points (a -ft, yi), (a, ^2). {a + h, 3/3), then 2ah?=yi + y3-2y2. 2. In the same case prove that 2ph^=4:y2a-yi{2a + h)-ys(2a~h), 2yh^=yi o (a + ft) + 2/3 a (a - ft) - 2j/2 (a^ - ft2). 3. Find the equations to the tangent and normal at any point {xi , yi) on the parabola whose equation is ky = x^. 4. Find the equation to the tangent at any point (xi, y{) on the hyperbola whose equation is y=- . Prove that, if the tangent at P meets X the axis of x in T and the axis of y in U, and if PN is the ordinate of P, N is the middle point of OT, and that P is the centre of a semicircle passing through T, O, U. (Fig. 17.) 41, 42] SOME APPLICATIONS OF DIFFERENTIATION 43 y T X Fig. 17. 5. Prove that, ity=^(r^-x^) oi y= -^(r^-x^), where ris independent dy _ X 'y' passes through the centre of the circle. oi X, ^ = - - ; and hence verify that the normal to a circle at any point Appeoximations 42. If we think what such a result as — ^^ — '- = nx'^~^ means. dx we remember that it implies that (X + hf - 03" . h is nearly equal to na;""^ when h is very small. It is convenient to have a symbol for "is nearly equal to" and we shall use the symbol " = ". Thus we write {x + Uy - a;" = ?ia!"-'A, or again (sc + hY = X* + nx^-%. 44 CALCULUS [CH. Ill If in this equation we put a; = 1 we get (1 +/i)"= 1+nh (1) This approximate equation is often useful. It gives an approximate value of the jith power of any number which diflFers very little from 1. For the approximation to be a good one it is necessary that n should not be too great. 43. The result (1) of § 42 can often be applied to the approximate extraction of square, cube, and other roots. As an example we find the fifth root of 31. ^ We have (31)* = (32-l)l = {32(l-i)} Hence (31)^=^,or(31)* = l-9875. The actual value correct to 5 places is 1 •98734. 44. In another way of using the approximate equation (1) of § 42 we suppose that x times some unit is the measure of a variable quantity, and .-e" times some unit is the measure of a related quantity. If the first quantity can be measured accurately to a units, where a is a small fraction, the true value of the first quantity may be anything between x + a units and x — a units. The true value of the, related quantity may be anything between (as + a)" units and (as - a)" units, or k" ( 1 + - j and x'' (1 — j units. Hence the possible error of measurement of the related quantity is approximately nax^~^ times the appropriate unit. 42-47] SOME APPLICATIONS OF DIFFKEENTIATION 45 45. Again the process of differentiating a product gives the approximate equation A (wy) = uAm + mA'U. This result may be illustrated by a figure. The area of a rectangle of sides u inches and v inches is uv square inches. If small additions Ait inches and ^v inches are made to the sides the area is increased by ©Am + mAd square inches approximately. In Fig. 18 the lengths of AB, AD can be taken to be u inches and V inches, and the lengths of BE, DG, Am inches and Ai; inches. The area of the rect- angle DGKC is mAi) square inches, and the area of the rectangle BEHC is ■wAm square inches. The actual increment of area differs from that calculated by the approximate rule by the area of the rectangle KFHC, which is much smaller than either DGKC or BEHC. 46. More generally, since the limit to which f{x + h)-f{x) h tends as h tends to zero is ./' (x), we have the approximate equation f{x + h)=f(x) + hf{x). 47. As illustrating this approximate equation we consider the coefficient of linear expansion and the coefficient of cubical expansion of a substance, e.g. copper. If a rod of the substance is warmed it becomes longer. Let the initial length be I inches, and the initial temperature t degrees Centi- grade. When f is changed to f -(- At, Z is changed to i -I- AZ, and, as AS tends to zero, — tends to a, limit, which is -^ . Let ^=a.l. Then a is called the At dt dt " coefficient of linear expansion," and we have the approximate equation l + M=l{i. + aAt). 46 CALCULUS [CH. Ill We see that, if At=l, so that the temperature rises 1 degree Centigrade, a is nearly equal to the ratio of the increase of length to the original length. If a Inmp of the substance, of yolome v cubic inches, is warmed, so that its temperature rises from t degrees Centigrade to t + At degrees Centigrade, V will be changed to v + Av, and, as At tends to zero, -r- tends to a limit, Hi? fil3 which is -J- . Let — =/3d. Then j3 is called the " coefficient of cubical expansion," and we have the approximate equation v + Av=v{l + pAt). As before, if the temperature is raised 1 degree Centigrade, /3 is nearly equal to the ratio of the increase of volume to the original volume. Now if the lump is a cube, of side { inches, v = l^, and dt dt' or pP = 3aJ3, so that ;8 = 3a. We have the result that the coefficient of cubical expansion is 3 times the coefficient of linear expansion. Examples 1. Find approximately by the method of § 43 the values of and (80)i. 2. The side of a cube can be measured accurately to -i^ of an inch, and the side is measured as 10 inches. Find approximately the possible error of measurement of the volume. 3. The sides of a rectangle can be measured accurately to rrjrrr of an inch, and the perimeter is measured as 125 inches, find approximately the possible error of measurement of the area. 4. The coefficient of linear expansion of copper is '0000167. By how much is a copper rod, 1 foot in length at 0° Centigrade, extended when its temperature is raised 10° Centigrade ? If the area of the section of the rod at the lower temperature is 1 square inch, what is its area at the higher temperature ? Maxima and Minima 48. As an example of a different kind of application of the Differential Calculus we take the problem of finding the shortest possible perimeter of a rectangle of given area. 47-49] SOME APPLICATIONS OF DIFFERENTIATION 47 Let the area be a square inches, and let y inches be the length of the perimeter, and x inches the length of one side. The length of the opposite side also is x inches, and the lengths of the other two sides are each {-zy — x\ inches. We have y Now % ax Hence, if x^ a, -~ is positive, the graph of y as a function of x goes up to the right. If we begin with any small value of x, and let x gradually increase, y at first diminishes, and it goes on diminishing until x^ = a, or x = Ja ; then y begins to increase, and it goes on increasing however great we make x. It follows that y has its least value when X = Ja. This least value is found, by substituting ija for x in the expression 2 / x+ - j , to be iJa. We see that the rectangle of given area and shortest perimeter is the square which has the given area. 49. We consider another problem of this kind. Suppose that we have a piece of cardboard in the form of a square of side 1 foot, and that we propose to make a box without a lid by turning up the edge of the square all round. The problem is to make the box of greatest volume. The full lines in Fig. 19 show the edges of the square, and the dotted lines show lines along which the cardboard may be bent. We take the height of the box to be x feet, (a; < 1). We see that we get a box with a square base of side 48 CALCULUS [CH. Ill 1 — 2a; feet and a height x feet, and, if the volume of the box is y cubic feet, y is given by the equation y = {\-2xfx. xl j a; X \ i^ \ X x\ Fig. 19. If we take x very small, we get a very shallow box and its volume is very small ; if we take x nearly equal to jj , we get a , deep slender box, and the volume is again very small. Now (Ex. iii in § 30) | = (l-2.)(l-6.) = 12g-.)(|-.). When X is very small, both the factors on the right are positive, J- is positive, and as x increases up to -^, y increases. When x lies between -^ and -^ the first factor is positive and the second negative, -^ is negative, and as x increases from ^ to g, j/ diminishes. The greatest value of y occurs when as = ? , and this 2 greatest value of y is ^ . The box of greatest volume is 2 inches Jit deep and its volume is 128 cubic inches. 50. These problems lead us to some general considerations concerning the application of the Differential Calculus to questions of maxima and minim,a. If y =J'(x), then so long as /' {x) is 49-52] SOME APPLICATIONS OF DIFFERENTIATION 49 positive, y increases as x increases ; when f (x) is negative, y diminishes as x increases. At a point on a graph, where y changes from increasing to diminishing, or from diminishing to increasing, /' (a;) vanishes. At such a, point the gradient of the graph is zero, and the tangent to the graph is parallel to the axis of X. When y changes from increasing to diminishing it has a Fig. 20. maximum value. When it changes from diminishing to increasing it has a m,inimum value. We can find the values of x which correspond to the maximum and minimum values of y if we can solve the equation /' {x) = 0, and we can find the maximum and minimum values of y by substituting in f{x) the numbers that satisfy this equation. 51. It may happen, as in the problem of the box, that the equation /' (cc) = is satisfied by more than one value of x, and then we have to choose the right value. In simple problems, such as we are likely to meet with, we can always do this. For example in the problem of the box we see that the equation is satisfied by a; =^ as well as by a = 5^. But we saw that - was the 2 6 right value, because when «^<-s> 2/ is increasing, and when a; > ^ , y is diminishing. We might have settled the point also by observing that when a; = ^ , 3/ = 0, and zero volume cannot be the maximum in such a problem. 52. The point can often be settled by using the second differential coefficient (§ 31). L. C. 4 50 CALCULUS [CH. Ill We know that, when f (x) is positive, /(x) increases as x increases, and when /' (x) is negative, /(x) diminishes as x increases. We know also that, if f{x) has a maximum or a minimum value, /' (x) vanishes for the corresponding value of x. If f" {x) is positive when /' (x) vanishes, /' (x) is increasing with X -f- and, therefore, in passing through zero, f {x) passes from negative values to positive values. Let as be a value of x for which f (x) vanishes and /" (a;) is positive. When a: is a little less than a, /' (x) is negative, and_/'(a;) diminishes as x increases towards a. When x is a little greater than a, f (x) is positive, and f(x) increases as x increases above a. Hence, as x increases through a,f{x) changes from diminishing to increasing. Therefore o is a value of x for which _/'(a;) is a minimum. In like manner, if 6 is a value of x for which f (x) vanishes and /" (x) is negative, it is a value for which f{x) is a maximum. For example, in the problem of § 49 f{x)=x(l-2x)% f'{x) = l-8x + 12x^, f" {x)=-8{l-3x). In thiB case /' (it) vanishes if i = j; or - . When x=-:, f" (x) is negative, and when x=^, /" (x) is positive. Henoea; = jT makes/(a;) amaximum,and X = g makes / (x) a minimum. 53. The length of the normal drawn from a point to a curve affords a good example of maxima and minima. Let P be a point on a curve, N a point on the normal at P. With centre N and radius N P describe a circle. Join N to Q a point of the curve near to P. If the part of the curve which is near to P is concave to N but outside the circle, N Q is greater than N P, and then N P is a straight line of minimum length drawn from the point N to the curve (Fig. 21 a). If the part of the curve which is near to P is concave to N but within the circle, N Q is less than N P, and N P is a straight line of maximum length drawn from N to the curve (Pig. 21/3). If the part of the curve which is near to P is convex to N, NP is a straight line of minimum length drawn from N to the curve (Fig. 21 y). 52, 53] SOME APPLICATIONS OF DIFFEEENTIATION 51 Fig. 21. Examples 1. The perimeter of a rectangle is given. Find its shape so that its area may be as great as possible. 2. The length of the diagonal of a rectangle is given. Find its shape in order that (i) its perimeter, (ii) its area, may be as great as possible. 3. Solve the problem of the box (§ 49) when the piece of cardboard has the shape of a rectangle of sides 13 inches and 15 inches. [Besult, depth = 2 '3155 inches approximately.] 4. Solve the problem of the box when the sides of the rectangle are a inches and 6 inches. 5. According to the regulations of the Parcel Post the largest parcels that may be sent have such dimensions that the sum of the length and the girth does not exceed 6 feet. Find the dimensions of the largest parcel in the shape (i) of a prism on a square base, (ii) of a cylinder on a circular base. [Besult, in both cases, length =2 feet.] Find the volumes of both parcels. 6. Determine the shape of a right circular cone of given volume in order that its surface may be the least possible. [Besult, height of cone : radius of base = 1-4142 approximately. ] 4—2 52 CALCULUS [CH. Ill The Theorem of Intermediate Value 54. If a function /(a) vanishes when x has the value a and also when x has the value h, the derived function f {x) must vanish for some value of x between a and h. This result is illustrated in Figs. 22 a and 22 yS. A B a? P Fig. 22. In Fig. 22 a the graph of f{x) goes above the axis of x between A and B, and f{x) has a maximum value at some inter- mediate point P. In Fig. 22 y8 the graph of f{x) goes below the axis of x between A and B, and /{x) has a minimum value at some intermediate point P. The function f{x) may, of course, have more than one maximum or minimum between x = a and x = h. It is certain that it has at least one. 55. This result is of very great importance in the more advanced portions of the Difi'erential Calculus. We may give it a slightly more general form by remarking that, if A and B are two points on the graph of a function, there is on the graph between A and B a point P at which the tangent to "^^^^^^ _. the graph is parallel to AB (Fig. 23). 56. This second form of the result is reducible to the first form. If the graph is that of f{x), and the gradient of the secant AB is m, we have b-a ' where a and h are the a!-coordinates of A and B. Now write 54-57] SOME APPLICATIONS OF DIFFERENTIATION 53 F (x) for the function f(b)-f (x) ~{b-x) m. Then F (a;) vanishes when x = 'b. Also, by the definition , of m, F {x) vanishes when x = a. Hence F' (a;) vanishes for some value of x between a and h. But F' («) = -/' (k) + m. Therefore/' (a:) = m for some value of x between a and h. We have the result^ that for some value of x between a and h. 57. The result may also be written where x is some number between x and x + h. The formula f{x + h)-f{x) ._ noted in § 46 is an approximate equivalent of the above exact equation. Examples X. Find the value of x which satisfies the equation f{b)-f{a) = {b-a)f'{x), (i) wheu/(x)=a;2, (ii) when/(a!)=a^. 2. Interpret the equation f{b)-f(a) = {b-a)f'{x) as showing that the average velocity of a moving body in any interval is the same as the velocity at some instant during the interval. If the body moves over s feet in t seconds, and s = at'^, where o, is constant, the instant in question is the middle instant of the interval. If s=/3«s, where ^ is constant, the instant in question is always later than the middle instant of the interval. Prove these statements. 1 For the necessary limitation of the result see Appendix II. 54 CALCULUS [CH. Ill 3. The theorem that, if /(a) vanishes when a: = o and when x=b, f {x) vanishes for an intermediate value of x, may sometimes be applied to determine the number and situation of the real roots of an equation. For example the equation 2a;3 + 3a;2+ 6x-10=0 has only one real root. Prove this. 4. Prove that the equation x^+px^ + qx + r = 0, in which p, q, r are independent of x, cannot have more than one real root itp^'cSq. CHAPTEK IV INTEGRATION 58. We consider some known results in regard to mensu- ration. (a) If the length of one side of a triangle is b units of length, and the length of the perpendicular let fall upon this side from the opposite vertex is p units of length, the area of the triangle is ^ bp units of area. Consider a right-angled triangle OAB (Fig. 24). Let the lengths of OB, AB be p, b units of length. Take O as origin and the axis of x along OB. Let x, y be the coordinates of any 56 CALCULUS [CH. IV point P on OA, let PN be the ordinate of P, and let the area of the triangle OPN be 2 units of area. Then 1 Since the triangles OAB, OPN are similar we have X p ' and therefore 2p This equation expresses « as a function of x. Since b and p are independent of x, we have dz b — = -x = y. dx p When P is moved along OA to Q, so that x is changed to x + Ax, y becomes y + Ay, and z becomes z + Az. The area of the trapezium PN M Q is Az units of area, and we have , 16, .,,16, A2=--(x + Ax)2-- -x2 Now and therefore Hence This is the known result that the area of the trapezium PNMQ is equal to that of a rectangle whose sides are MN and half the sum of PN and QM. We observe that Az lies between yAx and (y + Ay) Ax. (6) Let the radius of a circle be r units of length. Then the length of the circumference is 2Tir units of length, vfhere ir is = -Ax x + 5 p \ 2 AxV y + Ay _b _ x + Ax p y X ' Ay=- Ax. Az = Ax(y + l Aj/) =|{2/ + fe + A2^)}Ax. 58, 59] INTEGKATION a certain number which is approximately equal to 3 '14 16. The area of the circle is irfi units of area ^- Let then dz dr = 2Trr. Fig. 25. If the radii of two concentric circles (Pig. 25) are r and r + Ar units of length, the area of the figure contained between them is t {{r + Ar)^ - r^] units of area. When r is changed to r + Ar, z is changed to 2 + Ax, and As = 27rfr + -Ar j Ar, so that Az lies between 2irrAr and 2ir (r+Ar) Ar. The area of the included figure lies between the areas of two rectangles whose sides are, for one, the difference of the radii and the inner circumference, for the other, the difference of the radii and the outer circumference. 59. In the work that we have done so far ye have had to differentiate given functions, but in many applications of the ^ A discussion of the mensuration of the circle will be found in Appendix V. 58 CALCULUS [CH. IV Calculus we know the differential coefficient of a function before we know the function. We consider some examples. (a) Area under a curve. We consider a curve such as AB in Fig. 26, we take x, y to be the coordinates of a point P on the curve, and draw the ordinate PN. The area ACNP, bounded by AC, the axis of x, PN, and the arc AP, may be taken to be z Pig. 26. units of area. If we move P along the curve we change x, and also y, and we change z as well. We can regard z as a function of X. Usually we do not know what function z is, but we can find J. ax When P is moved to Q, N is moved to M, and we may express the length of N M as Aa: units of length. Also the length of QM is y + Ay units of length. The area ACNP is increased by the area PNMQ, and this area is Az units of area. Now, as the figure is drawn ^, this area is less than that of the rectangle QMNR, but greater than that of the rectangle PNMS, and the areas of these rectangles are respectively {y + Ay) Ax and yAx units of area. Az Hence Az lies between yAx and (y + Ay) Ax, and -— lies ^ In the figure the ordinate of Q is the greatest, and that of P is the smallest, in the arc PQ. The more general case is considered in Ch. T, § 75. 59J INTEGRATION 59 between y and y + Ay. Now when Aa; tends to zero, Ay also As tends to zero, and — — tends to a limit, which is y. But when Az -— tends to a limit, as Aa; tends to zero, that limit is the differ- Acc ential coefficient dz dx ' Hence we have the equation dz exactly as in § 58 {a). If we know the equation to the curve we can express y in dz terms of x, and therefore we know ^=- in terms of x. dx (6) Volume of a part of a sphere. We consider the volume of the portion of a sphere contained between two parallel planes, one plane passing through the centre of the sphere. We take the centre of the sphere as origin, and draw the axis of x at O H U Pig. 27. right angles to the planes. The surface of the sphere is generated by rotating a semicircle about its bounding diameter. Let P be any point on the semicircle (Fig. 27), we take it to the right of Oy ; and let PN be the ordinate of P. The section of the sphere by the plane which passes through Oy and is at right angles to Ox is a circle, and the section by the plane which passes through PN and is at right angles to Ox is another circle. Let x, y be the 60 CALCULUS [CH. IV coordinates of P, and let the volume of the portion contained between the two planes be v units of volume. Then « is a function of x. We do not know what function v is, but we can find^. ax When P is moved to Q on the semicircle, x becomes x + Ak, y becomes y + Ay, and v becomes v + Av. We note that when Ax is positive Ay is negative. The volume of the portion contained between the planes, which pass through PN and QM and are at right angles to Ox, is Av units of volume. Complete the rectangles PNMS, QMNR. As the figure rotates each of them traces out a slice of a cylinder. The cylinder traced out by PNMS stands ou a circular base whose area is iry^ units of area, and its height is Ax units of length. The cylinder traced out by QMNR stands on a circular base whose area is ■^ (y + Ayf units of area, and its height also is Aaj units of length. The volume Av units of volume lies between the volumes of these two cylinders. Remembering that Ay is negative we have Av < vy' Ax , but A« > ir (y + Ay)" Aa;, and therefore Av — lies between Try" and ir (y + Ay)^ Now, when Ax tends to zero, Ay also tends to zero, and — tends to a limit, which is -^ . ax Therefore -r- = tw^. ax If the radius of the sphere is a units of length we have x^ + y^= a^. Hence -=- = tt (a^ - x^), ax ^ ' so that we know -=- as a function of x. ax 59, 60] INTEGRATION 61 60. These examples are sufficient to show the importance of the problem : the differential coefficient of a function being given, it is required to find the function. In the first place let the differential coefficient of a function of X be zero for all values of x. Let y be the function. We have ^=0 ax for all values of x. If we had a graph of the function, the gradient of the graph would be zero everywhere. Now if y were increasing at any point, the gradient of the graph at that point would be positive ; if y were diminishing at any point, the gradient of the graph at that point would be negative. Hence y never increases and it never diminishes. Now the values of y at two points cannot be different without y either increasing or diminishing between the points. Hence the value of y is the same at every point. In other words y is the same for all values of x, it is independent of x, or we have the result that y is a constant. We can write. the result y = C, where C means a "number independent of x." The equation -^ = is satisfied by putting y=C, whatever value (independent of x) we give to C, and it cannot be satisfied in any other way. Since apy constant value may be given to C, it is often called an arbitrary constant. Next let the differential coefficient be 1. As before let y be the function. We have ,^ = 1 dx for all values of x. Now one way of satisfying this equation is to put y = x, but we cannot say that this is the only way. If possible let some other function y, different from x, satisfy the equation. We may write z for the difference y — x. Then we have y = x + z, and -^ = \, CALCULUS dx dz dz dx = 0. 62 CALCULUS [CH. IV but hence It follows that z must be a constant. As before it is. an arbitrary constant ; denoting it by C, we have y = x+ C. As a third example, we take the differential coefficient to be 2x. We have ^=2x ax for all values of x. The equation is satisfied by putting and, as before, the general solution is of the form y = as'+C, where C is an arbitrary constant. In the second and third examples we recognize the given value of -^ as a function which we have found to be the differ- ed ential coefficient of a particular function. Thus our problem is solved in two steps : first recognize the given differential coefficient as that of a particular function, and secondly add an arbitrary constant to this function. 61. When we recognize a function /{x) as the differential coefficient of some particular function (x) we are said to " integrate " the function /{x), and we call V(2 + 3=^). (4) x/(2-3a:), (6) W(l-.^), (7) ^^, In Ex. (1) — (5) it is better to work out the result in each case by making the appropriate substitution, in the way explained in § 66, than to write it down by substituting in the formula there obtained. It is the method, not the formula, that is important. Any one who has once grasped the method has no need of the formula. The results in all cases can be verified by differentiation, and it is therefore unnecessary to record them here. 5—2 CHAPTER V SOME APPLICATIONS OF INTEGRATION 69. We consider an example of the determination of the area under a curve (§ 59 a)). Let the curve be the graph oiy = a?, and let any straight line PQ be drawn across it parallel to the axis of x. Let {x, y) be Fig. 28. the coordinates of P. We can find the area bounded by the arc OP, the axis of x, and the ordinate PN. (Fig. 28.) If this is z units of area we know that Tx^y^"^- Hence z = :^a? + C, where C is some number independent of x. This result holds for all values of x. But if we bring P along the curve to O we diminish x to zero and we also diminish z to zero. Hence C must be zero, and we have z = \^. 69-71] SOME APPLICATIONS OF INTEGRATION 69 70. Since the length of PN is x2 units of length when the length of ON is X units of length, we have the result that the area bounded by the arc OP, the straight line ON, and the straight line PN is 5 of the area of the rectangle ONPL. Hence the area bounded by the arc OP, the straight line 2 PL and the straight line OL is ^ of the area of the same rectangle. Hence o also the area bounded by the are QOP and the straight line PQ is 5 of the o area of the rectangle PQMN. 71. More generally, let AC and BD be two ordinates of a curve (Fig. 29), PN an intermediate ordinate, a the a;-coordinate of Aj h that of B, x, y the coordinates of P, Let the area contained between the curve, the ordinates AC, BD, and the axis of a; be S units of area, and let the area contained between the curve, the ordinates AC, PN, and the axis of a; be s units of area. As in § 59 (a) we have dz dx -y- Let y =/{x) be the equation to the curve. Then we have 2= jf(x) dx + C, 70 CALCULUS [CH. V where C is independent of x. Let \f(x) dx= F {x), so that F {x) is an integral of f{x). Then » = F (a;) + C. Now as P moves backwards along the curve towards A, z tends to zero, or we must have F (a) + C = or C = - F (o). Hence » = F (a;) — F (a). Also S is the value of z when x = h, and therefore S = F (6) - F (a). It is to be observed that we determine the curvilinear area ACD8 by regarding it as a particular value taken by the curvi- linear area ACNP as P moves along the curve from A. 72. In § 71 we showed how to determine the value of z when x = h from the conditions : (^) £-/(^)' (ii) « = when x= a. The solution is found by first finding an integral F (x) oif{x). The arbitrary constant, which may be added to the integral F {x), or I f{x) dx, is determined by the condition (ii). When this is done the function z is determined as F (aj) - F (a). There is nothing arbitrary in the expression for z. The required value is then found by substituting h for x. The relation of the required value of « to the indefinite integral, and to the two particular values a and h of x, is indicated by writing the required value in rT> the form | f{x) dx. This expression is called the " definite Ja integral of f(x) with respect to x between the lower limit a and the upper limit 6." It means the value when 35 = 6 of a function which vanishes when x = a and has its differential coefficient with respect to x equal to f{x) for all values of x between a and 6 (inclusive). 71-73] SOME APPLICATIONS OF INTEGRATION 71 By contradistinction any integral of f{x), or \f{x)dx, is often called an "indefinite integral" ol/(x). Examples Find the values of the following definite integrals (1) — (8) : — (1) jxdx, (2) jx^dx, (3) r{x^ + x)dx, (4) jxdx, (5) I x^dx, (6) \\x^-x)dx, (7) \ ^dx, (8) / ~dx. J -1 jo J IX' J 0-01 '"'' The results are « I' (^)J. (B) ¥• (4) 0, (5) l (6) -I. (7) l (8) 99. 73. The following example is important in connexion with the approximate evaluation of integrals '. Let P, R be two points of a parabola whose axis is parallel to the axis of y, PL, RN their ordinates, M the middle point of LN, MQthe ordinate of a point Q on the parabola. (Fig. 30.) We may take the coordinates of the points P, Q, R to be : for P, a-h,yi; for Q, u, y^ ; for R, a + ft, j/s . If the equation to the parabola is y = ax^ + Px + y, 1 See Ch. X below. Hence 72 CALCULUS [CH. V and the number of units of area in the figure bounded by the arc PR, the ordinates PL, RN and the axis of x is S, we have fa+h J a-h S = ^a{a + h)3-{a-h)^ + ^P{{a + h)^-{a-h)2} + y{{a + h)-{a-h)} Now oa2 + |3a + 7=j/2, and yi + y3-2y2=«'{{a-hf+la + h}^-2a^ + ^{a-h + a + h-2a) = 2ahK (Of. Ex. 1, p. 42.) Therefore S = 2%2 + -71(3/1 + ^3 -2^2) 74. Let the area of the trapezium PLNR be T units of area. Then T = 2 2 A (!/i + 2/3) = ft (2/1 + 2/3), and T-S = Jft(!/i + 2/3-2j,2) = |2ft|^i±^^-2,2J.. Also the coordinates of V, the middle point of PR, are a, MlXM ^ and the length of VQ ia J^^ - Vi "iiits of length. The area of the segment of the parabola bounded by the arc PQR and the chord PR is T - S units of area. 2 Hence the area of this segment is - of that of a rectangle whose sides are LN and VQ. The result in regard to the area of the parabolic segment is proved here for the case where a is positive or the curve is concave upwards. It is un- altered if the curve is concave downwards. In this case 2 S-T=-ft(22/2-2/i-|/3)- 73-75] SOME APPLICATIONS OP INTEGRATION 73 75. In general, when we wish to find the area of a figure, we may proceed as follows : — Let any straight line drawn across the figure parallel to the axis of y meet the boundary of the figure in two points Pi and P^. Let y^ and y^ be the y-coordinates of these points, and let the suffix 1 be attached always to the upper point, the suffix 2 to the lower. Then yi>y2. We write Y for 2/i — 2/2. Let the straight line in question meet any straight line parallel to the axis of x in N. There will be two extreme positions of N, such as A, B in Fig. 31, and the whole figure will A N IVI B Fig. 31. lie between two straight lines drawn through A and B parallel to the axis of y. Let as, x, b be the cc-coordinates of A, N, B. We imagine N to move along the straight line AB from A to B. Then Y is a function of x, %a,j f{x). When N moves to M, so that x becomes x + Ax, Y will become Y + AY, where AY may be positive or negative. The values taken by /{x) when the upright straight line passes through a point between M and N need not lie between Y and Y + AY. The greatest of them may be a little greater than either Y or Y + AY 74 CALCULUS [CH. V and the smallest of them a little less than either Y or Y + AY. Call the greatest of them K and the smallest k. As Ate tends to zero, K and k tend to the same limit, viz. Y. Let the area of the part of the figure to the left of the upright line drawn through N be 21 units of area. When x is changed to a; ■)- Aa;, z is changed to a + As, and the area of the strip of the figure between the upright lines drawn through N and M is A« units of area. This area is less than that of a rectangle the lengths of whose sides are K and Aa; units of length, but it is greater than that of a rectangle the lengths of whose sides are k and Aa; units of length. Hence As lies between KAa; Az and AAa;, and — lies between K and k. Ax Az As Aa; tends to zero, -:— tends to a limit which is Y, and we Aa; have fife Let the area of the figure be S units of area. Then S is the value of z when x= h, and the value of z when x = a is 0. Hence S = I (2/1 - Vi) dx. Ja It may be observed that this discnssion applies to the area under a curve [§ 59 (a)]. In this case j/2=0, and y may be written for yi. Examples 1. We may use the method of § 75 to find the area of a triangle. In each of the figures (Figs. 32, 33) we take the lengths of AB, OD, ON, PiPj to be 6, p, X, Y units of length, and we have — = - . Hence S = I - xdx = ■= op. joP 2 ^ 2. Prove that for a segment of a circle, of radius r units of length, cut off by a chord distant a units of length from the centre, S= iZ^ir^-x^dx. J a [The indefinite integral will be evaluated in Oh. VIII.] 75,76] SOME APPLICATIONS OF INTEGRATION 75 3. Draw roughly the graph of y=x (1-x), and find the area contained between the curve and the axis of x. [Eeault, ^ of a unit of area.] 4. Do the same iovy=x(l- 2x)2. [Result, jo of a unit of area.] 5. Draw roughly the graph of x(l-x)(2-x), and find the two areas enclosed by the curve and the axis of x. [Result, 7 of a unit of area in both Fig. 33. Volumes of Solids 76. We found in g 59 (h) that, if the distance between a plane drawn through the centre of a sphere and a parallel plane is X units of length, and the volume of the included solid is V units of volume, where the radius of the sphere is a units of length. Also we have v=<) when a; = 0. Hence v = it i a^x — ^ a;^ j . If we put a; = a, we have the volume of a hemisphere, viz. 76 CALCULUS [CH. V ^ira' units of volume. The volume of the complete sphere is ira^ units of volume. 77. The same reasoning will give us the volume of a cone. Let O be the vertex of the cone, let its height be a units of length, and the radius of its base b units of length. Take the axis of x along the axis of the cone. Through any point N on the axis let there pass a plane at right angles to the axis. The section of the Fig. 34. cone by the plane is a circle. Let PN be its radius, x, y the coordinates of P. The area of the circle is iry^ units of area. Let the volume of the part of the cone included between this circle and the vertex be v units of volume. When x becomes X + Aa;, V becomes v + Ad, and just as in § 59 (6) we find that Av is intermediate between Try^Aa; and Tr(y -v h.yf^x. dv Hence But we have dx X 76-79] SOME APPLICATIONS OF INTEGRATION 77 and therefore -r- =-n-—„sc'; ax (t also « = when ai = 0. Hence 'U = -Tr - a?. 3 a^ If the volume of the cone is V units of volume, V is the value of V when x = a, or we have The volume of the cone is ^ of the volume of a cylinder whose o height is the height of the cone and whose base is the base of the cone. 78. The same method may be used to find the volume of any solid of revolution. Let the curved bounding surface of the solid be generated by rotating a curve about an axis, which we take to be the axis of x. Let a fixed plane cut this axis at right angles, and let a plane passing through any point (x, y) on the curve also cut the axis of x at right angles. Let the volume of the portion of the solid contained between the two planes be ■w units of volume. Then vis a function of x, and -=-= Try'^. 79. This method can be generalized so as to apply to any solid. Let a fixed plane and a variable plane both cut the axis of X at right angles, and let x be the ^-coordinate of the point where the variable plane cuts this axis ; also let the area of the section of the solid by the variable plane be Z units of area, and the volume of the portion of the solid contained between the two planes be v units of volume. Then Z and v are functions of x and we have -^ = Z. If we know how to express Z as a function of X and to integrate this function we can find v. 78 CALCULUS [CH. V 80. As an example consider the volume of » pyramid on a triangular base. Let the area of the base be B units of area, and the length of the perpendicular O D let fall from the vertex O upon the plane of the base be p units of length. Take O as origin and the axis of x along OD. (Fig. 35.) Through any point N on the axis of x let there pass a plane at right angles to this axis cutting the edges OA, OB, OC of the pyramid in P, Q, R. Fig. 35. The area of the triangle PQR is what we have called Z units of area when the length of ON is a: units of length. The volume of the pyramid with vertex O and base PQR is what we have called v units of volume ; and we have the equation dv _ dx" Now the triangles PQR, ABC are similar, and the areas of similar figures are proportional to the areas of the squares described on corresponding sides. The ratio of a pair of corresponding sides is the same as that of any two corresponding lines in the two figures, e.g. the lines PN, AD or ON, OD. Hence we have • Z xi B~y2- and also » = = when x= = 0, and therefore ''=3 ^2^'- 80, 81] SOME APPLICATIONS OF INTEGRATION 79 If the Tolume of the pyramid is V units of volume, we find, by putting x=p, that or the volume of the pyramid is s of the volume of a prism of the same height and base. Nothing in the argument depends upon the base having three sides, and the result holds for any pyramid. Examples 1. Prove that the volume of a frustum of a pyramid or cone is -{A + o + ;v/(Aa)}ft units of volume, the areas of the two parallel faces being A and a units of area, and the distance between these faces being h units of length, 3. An arc of the parabola y = ^x, terminated by the origin and a given point, revolves about the axis of x. Prove that the volume of the solid bounded by the surface traced out, and by the plane which passes through the given point and is at right angles to the axis of x, is one-half the volume of a cylinder of the same base and height. 3. An arc of the parabola y = x', terminated by the origin and a given point, revolves about the axis of x. Prove that the volume of the solid bounded as in Ex. 2 is one-fifth of the volume of a cylinder of the same base and height. Uniformly accelerated motion 81. When a body, such as a falling stone, moves without rotation, and without change of size or shape, the motion of any point of it is the same as the motion of any other point of it. The coordinates of one point of the body at any instant specify the position of the body at that instant. If every point of the moving body moves parallel to a fixed straight line we may speak of the body as moving in a straight line. If the straight line is the axis of x, the position of the body at any instant is specified by the jc-coordinate of one point of it. If the unit of length is a foot, the point in question is 80 CALCULUS [CH. V X feet to the right of the origin if x is positive, x feet to the left if X is negative. Iiet the position of the body be specified by x at the instant which is t seconds later than some chosen instant. If the point moves in the sense of increase of x the body has a velocity of -=- feet per second in this sense. If the point moves in the opposite sense the velocity is —-it feet per second. In both cases -J- is the measure in foot-second units of a certain quantity called the "velocity in the direction of increase of x." If the velocity is variable, the second differential coefficient -^, if positive, measures, in foot-second units, the rate per second at doc cl/3G which the velocity measured by -5- increases. If -5-j is negative (Px — -j-^ measures, in foot-second units, the rate per second at which the velocity measured by -=- diminishes. In both cases ~ is the Cltt Cbl measure in foot-second units of a certain quantity called the " acceleration in the direction of increase of x." (Cf. § 32.) If the line of motion is the axis of y we have only to write y instead of x. 82. An unsupported body near the Earth's surface would, if the air offered no resistance, fall with a constant acceleration. This acceleration is called the " acceleration due to gravity." We generally write g for the numerical measure of this acceleration, meaning that this acceleration is g units of acceleration. In foot-second units g is 32 '2. We set up a system of coordinate axes so that the axis of y is the vertical at a place drawn upwards. Let the position of the body at the instant which is t seconds later than some chosen instant be specified by the coordinates x, y of one point of it. If 81-83] SOME APPLICATIONS OF INTEGRATION 81 the body is free to move and we neglect the resistance of the air we have the equation If the body is let fall at the instant from which t is reckoned, and if the value of y at that instant is y^, we have 2/ = ^0 and -^ = when t=Q. Write V for -^ . Then we have dt dv -^ = ^ and v = when t = 0. Hence v = -gt. This is ^y - nf which gives 1 '+c. To make y = y„ when i = we must have Hence y = y<^--^0^- A.'D.^ point of the body is -^ gt^ feet lower down at the instant specified by * than it was at starting. The velocity of the body at this instant is gt feet per second in the downwards direction. 83. If the body instead of being let fall, is projected in some direction, we take the axis of x to be in the vertical plane passing through the direction of projection of one point of the body. Then every point of the body moves in a plane parallel to this plane. If, as before, we neglect the resistance of the air, L. c. 6 82 CALCULUS [CH. V any point of the body has an acceleration of g units downwards and no horizontal acceleration. "We have the equations d^x _ _ (Py _ ■^~ ' d^ ^^ At starting the body has some horizontal velocity and some vertical velocity. "We may suppose that -^ = M and ~ = v, when t = Q. dt dt ' Then u and v are independent of t. The horizontal velocity at starting is u feet per second and the vertical velocity at starting is v feet per second, upwards. "We may take the origin at the point of projection, so that x = and y — Q when < = 0. ' ft 'V fl'V The equation -z-^ = shows that -j- is some constant; and the Ctt Orb condition -v- = m when t = Q then shows that -=- = ■" always. The dt dt •' horizontal velocity is always the same as it is at first. The equation -=- = u, shows that x = ut + C where C is constant, and the condition a; = when t = shows that C = or x = ut. The equation --A= — 9 shows that -^ = —gt + /^, where A is dt ctt constant, and the condition -f = v when * = shows that A = v. dt Hence -—■ = v —gt. This equation shows that y = vt — -^g^+B, at ^ where B is constant, and the condition y = when t = shows that B = 0, or y = vt-^gt\ 83] SOME APPLICATIONS OF INTEGRATION 83 We may eliminate t between the two equations x = ut and 1 2' ^ 1v? u y = vt~j.gf. We get The form of this equation shows (§35) that the path of the mo^dng point is a parabola with its axis vertical, and that the curve lies below the tangent at the vertex (Fig. 36). Fig. 36. Examples 1. A body moves in a straight line with a constaat acceleration / foot- second units. Prove that, if its velocity is u feet per second in the sense of increase of s at the instant from which t is reckoned, and v feet per second at the end of t seconds, »=ii+/t. Prove that, if the body passes over s feet in t seconds, s=Mt + ^/t2. 2. In the notation of Ex. 1, — =/ and v=- -z- , so that v -j- =f-r . dt at Qit u/t Hence prove that v^-u^=2fs. — I - 2/s = const. is satisfied, / being a constant, prove that the body moves with a constant acceleration. 6—2 CHAPTER VI LOGARITHMS AND THE EXPONENTIAL FUNCTION 84. We have as a definition.^ of logarithms to base 10 (common logarithms) the statement that, if x= 10", then y = logio x. This definition would enable us to construct for ourselves a table of logarithms. We should begin by finding ^10 by the ordinary arithmetical method correctly to a number of places of decimals. To 16 places we have 10*= 3-1622776601683793. From this we can find 10* correctly to 8 places (1-77827941), then 10* correctly to 4 places (1-3335), then 10" correctly to 2 places (1-15). In the same way we can find by purely arithmetical methods the values of 10^^ 10^^ 10', 10^^ 10* 10^ 10^, 10* 10 **, 10^, 10**. If then we take a; = 10" we have the values of x for a number of values of y. We can then, put down the following table : 1 A discussion of the definition will be found in Appendix III. 84] LOGARITHMS AND THE EXPONENTIAL FUNCTION 85 Table I. y tV 4 A i A f A i X 1 1-15 1-33 1-54 1-78 205 2-37 2-74 3 16 y A f ik * H i 11 1 X 3-65 4-22 4-87 5-62 6-49 7-50 8-66 10 In this table all the values of x are correct to 2 places of decimals. From this table we can draw a graph of y = logio x between a; = 1 and a; = 10, and by means of the graph we can read off the logarithm of any number between 1 and 10. If the work is well done the value we can read off will certainly be correct to 1 place and often to 2. By means of the law of indices the table, or the graph, can be extended to values of x which do not lie between and 1. y_ I 3/=logiox Fig. 37. The graph of \ogy,x between a; =0-1 and a; =2 is shown in Fig. 37. "We see that anyone who took trouble enough, and made no 86 CALCULUS [CH. VI mistakes, could construct a table of logarithms, correct to as ■ many decimal places as he might wish, by merely finding square roots, just as a business man might make his own ready reckoner. But it saves time to buy one. For a similar reason we buy a Table of Logarithms^. 85. In working with logarithms the moat important fonunlcB are log (ab) = log o + log b, log m= logo - log 6, log (a") = n log a. These hold for any base. In regard to change of the base the chief formulee are log„!i; = (log6a;)(log„i), (loga6)x{logja) = l. 86. We consider logioa;as a function of x, and seek to differ- entiate it. We have logio {x + h)- logio a: = logio ^^ = logi„ (l + -) > and = -!'*('*;)'• We wish to show that this expression tends to a limit when h tends to zero. We write n for t , and have /i logi,(a; + /i)-logjoa ; _ 1 ,^^ /, , 1 ^ - . ■ f 1\" Now it is not difficult to convince ourselves that ^ The tables of logarithms in books of tables are not constructed by finding square roots in the manner explained above, but by a different method depending upon the use of|infinite series. 84-86] LOGARITHMS AND THE EXPONENTIAL FUNCTION 87 tends to a limit i as n increases. We consider logio(l + -j , or n logio ( 1 + - ) . In Table II. we have the values which this expression takes as n increases from 1 to 9. We see that each value is a little greater than the one before it, but the differences get smaller. Table II. n 123456789 nlogio (l + -) 0-3010 0-8522 0-3748 0-3876 0-3959 0-4017 0-4059 0-4092 0-4118 Now we go a little faster, and put down in Table III. the values which the expression takes when n is equal to 10, 20, ... 90. We see that the logarithm always increases but the differences are getting much smaller. Table III. n 10 20 30 40 50 60 70 80 90 n login ('l +-") 0-4139 0-4238 0-4272 0-4290 0-4300 0-4307 0-4312 0-4316 0-4319 Again we will go faster still and put down in Table IV. the values which the expression takes when n is equal to 100, 200, ... 900. We see that the logarithm tends to a constant value. Table IV. n 100 200 300 400 500 600 700 800 900 nlogjo^l +i'\ 0-4321 0-4332 0-4336 0-4338 0-4339 0-4339 0-4340 0-4340 0-4341 88 CALCULUS [CH. VI If now we go further, and take n to be 1000, 2000, ... we find that all the values of the logarithm, correct to 3 places, are 0-434. With a seven-figure table we cannot be sure of more than three places. Hence we conclude that, as n increases, logu, ( 1 + - ) probably tends to a limit, which is approximately equal to 0-434, and that (1 + -) probably tends to a limit which is approxi- mately equal to 2-72. It can be proved formally that ( 1 + - ) tends to a limit as n increases^ This limit is a perfectly definite number denoted by e. The value of e correct to 4 places of decimals is 2-7183, and the value of logu, « correct to 4 places of decimals is 0-4343. We shall denote the number logjje by M. Now we have the result d logio X _ M cfo X 87. Instead of taking logarithms to base 10 we might take logarithms to any base a. We should have as a definition the statement that if a; = a" then y = log^ x. By the process adopted in § 86 we should find X loga (x + h)- \0ga{x) _ 1 / h\h 4'°«.(-;)*. and thence ^"^^^^. = hos^e = dx XX logj a ' In particular we might take e as base, and find d logj X 1 dx x' 1 See Appendix IV. 86-89] LOGARITHMS AND THE EXPONENTIAL FUNCTION 89 88. This result gives us a new standard form of integral, I- dx --= loge X. (B) J X The result may also be written dx= — log. ii X M 'Sio ^' where 1 = log, 10 = 2-3026. We can write down also by the method of §§ 66, 68 the more general results : dx= - log, (ax + h), 'ax + b a "'^ " I - ^- dx = log, z. I zdx *' h The Exponential Function 89. When y = log, x, x = e", and we know that dy^l dx x ' TT dx Hence -^ =x, dy or -\~! = ev. dy Here e" is regarded as a function of y. It is known as the " exponential function." If we write x in place of y, e^ is the exponential function of x, and we have A if) ,.. dx 90 CALCULUS [CH. VI In the following Table the values of ^ are given, correctly to 4 places of decimals, for a number of values of x lying between - 1 and 1. Values of e^ in other ranges of values of x can be deduced by help of the law of indices. X -1 -0-9 -0-8 -0-7 -0-6 -0-5 -0-4 -0-3 -0-2 -0-1 e» 0-3679 0-4066 0-4493 0-4966 0-5488 0-6065 0-6703 0-7408 0-8187 0-9048 1 X 0-1 0-2 0-3 0-4 0-5 0-6 0-7 08 09 1 e' i-1052 1-2214 1-3499 1-4918 1-6487 1-8221 2-0138 2-2255 2-4596 2-7183 The graph of e" between x = -\ and a; = 1 is shown in Fig. 38. IJ 1 I _J t- ni 7 2 r 7 :t 7 t I ^ r -^ T l7 / / _,Z .^ ^^ Z^'^ "0 -r Fig. 38. 89-92] LOGARITHMS AND THE EXPONENTIAL FUNCTION 91 90. If X is negative and numerically large e" is positive and small, and it diminishes rapidly as the negative value of x increases numerically. In other words, as the positive number x increases e~^ continually diminishes and rapidly assumes very small values. On the other hand, if the index x is positive, e'° continually increases with x, and rapidly assumes very large values. 91. If we apply the rule for differentiating a function of a function (§ 25) we find at once that dx a being any constant. This result gives us an important in- tegral, viz.. /■ e'^dx=- e"^. (C) In this formula a may be any positive or negative number. 92. The function a" is often called an exponential function. Since we have (i=^=e^l<'8e", and ^^=log,a.c»'<'!?e''=logea.a*. This gives us an important limit. If we attempted to differentiate a" directly we should form the quotient ft ' which is the same as w' — = — . h It follows that, as h tends to zero, —r — tends to a limit which is logja. 10* - 1 1 In particular — - — tends to the limit ^, approximately equal to 2-3026 ; jft _ 1 e* - 1 — = — tends to the limit zero, and — - — tends to the limit 1. The number h h e is distinguished from all other numbers by the fact that when a = c the a*- 1 . limit of — J — is 1 ; when a>e or a<.c the limit is not 1. 92 CALCULUS [CH. VI Examples 1. Differentiate the following (1)— (12) :— (1) log,{2 + x). (2) log. (1- 2a;), (3)ioge^:. (4) log,|±i, (5) xlogeX, (6) {log,(x)}2, (7) xe", (8) x2««, (9) e-", (10) e-\ (11) a-x)e'. (12) (l-a!)2e». 2. Provethat '?[l°ge{^H-y(.2 + C)n ^ 1 da; ^(x2 + C)' C being independent of x. Cf. Ex. (ii), p. 32. 3. Prove that if h is very small log. (1 + A) = ft, and logs(a; + ft)-logjX==-. 4. Prove that when ft is very small 6*== 1 + ft. 5. Integrate the following (1) — (6) : — « ih' (2) 1^' (^) x4i. (*) •'-^ (5) x«^\ (6) ^. Verify the results by differentiation. 8. Integrate _^^ and -^^ . 7. Prove that, \iy is a function of x, — ^^ — '- = e.'^\ -j-^ay I . 8. Prove that, if 771 is constant, the equation is satisfied by putting ^ = Ae'"*+ Be"™', where A and B are any constants. 9. Prove that ^ + a ^=.-i- pt^ _ 1 2 ei^-j/i , ax^ ax ( ax2 4 "J a being a constant and y a function of x. Hence prove that, if x satisfies the equation -^ + fc — +ra2x=0, xe*** satisfies the equation d2(xci**) ^n2-ift2Vxc4**) = 0, dt2 A and n being constants, and x a function of t. 92, 93] LOGAEITHMS AND THE EXPONENTIAL FUNCTION 93 Applications of the Exponential Function 93. We consider some problems which can be solved by means of the exponential function. In Physical Chemistry^ we meet frequently with problems in which a number x, representing the measure of concentration of a solution, is to be determined in terms of a number t, represent- ing the measure of the time elapsed since some instant, by an equation of the form -=- =f{x), where f{x) is a given function. For example, the rate of inversion of cane sugar at any instant during the process of inversion is proportional to the amount that has not been inverted at that instant. The amount is specified by the degree of concentration. If a measures the initial con- centration, X the concentration of invert sugar at the end of t minutes, we have the equation dx -^^=k{a-x), where A is a constant, which is found to have the value O'OOIS approximately. We have to solve this equation, and we must also satisfy the condition that a; = when t = Q. dt 1 We have dx kia — x)' or d (kt) _ 1 dx a — x' and therefore kt = / dx+C J a-x = -log,(a-x) + C. To make x = = when < = we must have C = logea, and therefore M = log, a - loge {a-x) = log, a a — x' See J. W. Mellor, Chemical Statics and Dynamics, London, 1894. 94 CALCULUS [CH. VI CALCULUS a e*', a — x a — x e-"', a 5=1- a -e-**. so that This formula gives the solution of the problem. 84. As another example, let the velocity of chemical change be given by an equation of the form -j- = A; (a-x){b-x), where k, a, b are constants, and let x=0 when {=0. We have to express x in terms of t. We shall suppose that b is the greater of the two numbers a and b. We have k r^ = ] dx (a-x)(b-x) ' and we observe that we know how to integrate the sum h , and a-x b-x also how to integrate the difference = , and further we observe a—x b-x that 1 1 l-a a-x b-x (a-x) {h — x)' Multiplying both sides of our equation by (6 - a), we have ft (6 - a) -J- = , . ' ax a-x b-x Hence k(b-a)t=i( )dx+C = - loge {a-x) + log, {b-x) + C 1 b-x ^ To make x=0 when t=0 we must have C=-log.^=log.^, and therefore k{b-a)t= logj ( —— . r j > 93-95] LOGARITHMS AND THE EXPONENTIAL FUNCTION 95 or il^z£) = eft(4-a)« b {a-x) ' so that a;(6«*(6-«)*-a) = a6(e*(*-»)*-l), and the solution of the problem is given by the equation x = ab ),gk(b-a)t_a- 95. In the theory of Electric Currents ^ we meet frequently with problems in which a number i, representing the measure of a current, is to be determined in terms of a number t, represent- ing the measure of an interval of time elapsed since some instant, by an equation of the form di where L and R are constants, representing the measures of the coefficient of self-induction and the resistance of a circuit, and E is a constant or a function of the time, representing the measure of an impressed electromotive force. We have to determine i so as to satisfy this equation and the condition that i~0 when t = 0. We take the case where E is a constant, and write '="■ Then di ,. E dt L N"ow (Ex. 7, P- 92) '-P"'0') and therefore dt ~L^' so that ie"'-- = ^^e'" + C = -e'" + C. Lb R 1 See J. J. Thomson, Elements of the Mathematical Theory of Electricity and Magnetism, Ch. xi, Cambridge, 1895. 96 CALCULUS [CH. VI To make i = when t= we must have E C=--, R' and therefore ie" = — (e*" - 1), R R ^ ' The solution of the problem is given by the equation This formula gives the intensity of the current produced in a circuit by a constant electromotive force, the circuit being closed at the instant specified by i! = 0. 96. We may take account of the resistance of the air to the motion of a falling body by assuming that this resistance is pro- portional to the velocity. This assumption gives a good approxi- mation to the effect of the air on the motion of a small body which is not moving very fast. Let the body fall through s feet in t seconds from the start. ds Its velocity at the instant specified by < is -i- feet per second, and its acceleration in the downwards direction is -^rs foot-second units of acceleration. The force of the earth's gravity gives it an acceleration g, or 32"2, foot-second units, in the downwards ds direction. The resistance of the air gives it an acceleration k -r. dt foot-second units in the upwards direction, k being a constant. We have therefore the equation dh , ds dfi^^'^dt' 95, 96] LOGARITHMS AND THE EXPONENTIAL FUNCTION 97 ds and the conditions that s = and -i- = when t = Q. The above dt equation is dh ds dt^^^dt- = 9'- As in § 95 we may write this equation in the form dt -.ge^\ so that ' dt = Ige'^dt + C ds To make -=- = dt ■■ when < = we must have and therefore -O. We see that, as t increases, — increases, but that it never txt becomes greater than j . The velocity tends to a limiting velocity I feet per second. This velocity is called the "terminal velocity." To find s in terms of t we have the equation dt'k^^ " >' and the condition that s = when t = 0. The equation gives where C' is a constant. To make s = when * = we must have L. c. 7 98 CALCULUS [CH. VI Hence , =| « _ |(1 _ e-*'). We see that if the body falls for a long time (so that t is large), the distance through which it falls is nearly the same as if it fell with the uniform velocity | feet per second for t--r seconds. Examples 1. In the problem of § 93, taking a = 10-023, x=0 when « = 0, ic=l-946 ■when t = 60, find k, and make a table of the values of x when J =30, 90, 120, 150, 180. 2. In the problem of § 94, taking a=10-023, 6=89-977, x=(i when f =0, a; = 1-946 when t=60, find h, and make a table of the values of x when t = 30, 90, 120, 1^0, 180. 3. In the problem of § 95, taking E=20, R = 2, L=4, find the value of i when t = l, 10, 60. Do the same, taking E=20, R = 4, L=2. 4. A tank is being emptied, in such a way that the rate at which the water is flowing out at any instant is proportional to the amount left in at that instant. If half the water flows out in five minutes, how much will flow out in 10, 15, 20 minutes ? 5. Eain is falling with a velocity of 60 feet per second. Assuming that this differs very little from the terminal velocity (§ 96), and that the resist- ance is proportional to the velocity, find approximately the retardation of the drops by the resistance, and prove that the time taken to acquire a velocity of 54 feet per second is 4-29 seconds nearly, also that the distance through which a drop falls in the first second of its fall is 14 feet nearly. CHAPTER VII TRIGONOMETRIC FUNCTIONS 97. If the radius of a circle is r units of length, and the length of any arc is s units of length, the number - is a measure of the angle which the arc subtends at the centre. If we put B 8 ... 1 for - , the angle is 6 radians-^. A right angle is ^ ir radians, or 2 1-5708 radians approximately, and a radian is — right angles or TT 0-6-366 right angles approximately. The area of the sector which stands on an arc is -^ r^O units of area when the angle subtended by the arc at the centre is Q radians, the radius being r units of length. 98. It is convenient to think of angles as capable of taking any magnitude, greater or less than two right angles. For this purpose we think of the angle as traced out by one of its sides revolving about the vertex of the angle. We may think of the revolving line as starting from the position occupied by one side of the angle, and turning round the vertex, until it comes into the position occupied by the other side. ' A discussion of the radian measure of angles will be found in Aj)- pendix V. 7-2 100 CALCULUS [CH. VII "We generally think of the vertex of the angle as the origin of a system of coordinates x, y, and of the starting position of the revolving line as the axis of x drawn towards the right from the origin. We also agree that, when 5 is a positive number, an angle of radians is traced out by a line revolving in the opposite direction to the hands of a watch, placed face upwards on the paper (Fig. 39). According to this convention the sides of an angle of 2nir + 6 radians (n being a positive integer) are in the same positions as the sides of an angle of 6 radians. Fig. 39. 99. Angles of magnitude greater than two right angles are important in connexion with the rotation of bodies. If we think, for instance, of a rotating fly-wheel, and of the motion of a line traced on one of its faces, we see that the line turns through 2ir radians in every complete revolution of the wheel. In any interval, say t seconds, the line will turn through some angle, say ff radians, and S may be greater or less than tt according to circum- stances. As the wheel turns, ff increases, and B is some function of t. If the wheel turns uniformly, S is a simple multiple of t, and - is the measure in radians per second of the ' ' angular velocity " of the wheel. Whether the wheel turns uniformly or not, -^ is the measure in radians per second of the angular velocity. 98-102] TRIGONOMETRIC FUNCTIONS 101 100. Let P be any point in the plane of the coordinate axes, and let OP be one side of an angle of which thte other side is Ox. Let the measure of the angle traced out by the revolving line as it turns from the position Ox to the position OP be ^ radians, and let the distance OP be r units of length. Let x, y be the coordi- X 1/ nates of P. Then the numbers - , - depend upon Q, but not on anything else. They are functions of 6, called the cosine and the sine. We write - = sin c*, - = cos &. Fig. 40. 101. Since a;^ + y^ = r^, we have the result which is usually written sin^ Q + cos^ 6=1, in which sin^ B means (sin 6)", and cos" B means (cos Sf. 102. In Fig. 40 the two lines marked POP are at right angles to each other, and all the triangles marked PON are equal 102 CALCULUS [OH. VII in all respects. Hence, attending to the signs of x, y we have cos5 = sinr^Tr + 6j , sin 5 = -cos (g'T + ^) ••■(^)- Further %m 6 = — sax {it + 6), cos 6 = - cos (tt + 6) (2), and sin 6 = sin (27r + 6), cos 6 = cos (27r + ^) (3). 103. In Fig. 41 the two lines marked OP, OP' are equally inclined to the axis of x. If the acute angle asOP is & radians the obtuse angle asOP' is ir - 6 radians. Attending to the signs of x, y we see that sin (it — 6) = sin 6, cos (ir — 6) = - cos 6 (4). Pig. 41. By means of these formulae we can write down the values of sin 6 and cos 6 for any positive value of 6 when the value of sin $ is known for values of 6 between and ^ ir. 104. It is sometimes important to observe that, any two numbers a and 6 being given, it is possible to find one, and only one, positive number r and, at the same time, one, and only one, number 0, between and 27r, so as to make a=rcoa6, h=rsva.e. Either or both of the numbers u, and b may be negative ; but, whether they are negative or positive, the two are coordinates of one, and only one, point, say P. This point is at a perfectly definite distance from the origin, and the numerical measure of this distance in terms of the unit of length is the 102-105] THIGONOMETKIC FUNCTIONS 103 required number r. The angle through which a straight line, starting from the position Ox, and revolving through less than one complete revolution in the counter-clockwise sense, would have to turn in order to come into the position OP is a perfectly definite angle, and the numerical measure of this angle in radians is the required number 9. 105. We might make our own table of sines, just as we could make our own table of logarithms. To see how this could be done we notice that if we draw an angle, e.g. j of a right angle as accurately as possible, we can mark a point P on one of its sides, draw a perpendicular from P to the other side, and measure as accurately as possible the length of the perpendicular and the distance of the point P from the vertex of the angle. Then the fraction number of units of length in the length of the perpendicular number of units of length in the distance is the sine of the angle, and the sine can therefore be determined with a degree of accuracy that depends only on the accuracy of our measurement. Just as in the case of logarithms, we do not actually calculate a table of sines, but buy one^ In an ordinary table of sines the angles are given in degrees and minutes (a minute is ^ of a degree, and a degree is ^rr of a right angle). We can reduce the measure of the angle to radians by the rule that 180 degrees is ir radians. By means of the ordinary tables we could construct a new table ^ in which the angle is expressed in radians, and the sine of each angle is given correctly to a few places of decimals. The following table gives a few corresponding values correctly to three places. ' The tables of sines in books of tables are not constructed by measuring lengths in the manner described, but by a different method depending upon the use of infinite series. 2 Tables of this kind have been constructed by C. Burrau. See his ' Tables of cosine and sine of real and imaginary angles expressed in radians.' Berlin, 1907. 104 CALCULUS [CH. VII e sin e 0-1 0-2 0-3 0-4 0-5 100 0-199 0-296 0-389 0-479 0-6 0-565 0-7 0-644 0-8 0-717 e sin 9 0-9 1 1-1 1-2 1-3 0-783 0-841 0-891 0-932 0-964 1-4 0-985 1-5 0-997 1-6 1 :iiiiii~e:^ii::i: 7 i 5 106. In what precedes 6 is a variable number and sin 5 is a function of that variable. We thought of 6 as the number of radians in the measure of some angle, but we need not think of it in that way. We may take the two numbers 6 and sin 6 to be simultaneous values of two variable numbers x and y, and then y is a certain function of x, the sine of x. We may use the table in § 105 to plot the graph of y = sin x when x lies between and ^TT (or 1-5708). The graph is shown in Fig. 42. We may now use the re- sults of §§ 102, 103 to continue the graph of sin x beyond x = -^Tr. The graph consists of a succession of exactly equal and similar bays lying alternately above and below the axis of x, as shown in Fig. 43. We may then use the result cos x = sin (^tt + x) to draw the graph oi y = cos x for positive values of x. The result is shown in Fig. 44 which is obtained by shifting the curve in Fig. 43 through -= TT units of length towards the left band. 107. To define sin x and cos x for negative values of x, we simply make it a rule that the whole of the graph in each case y = sva X. Fig. 42. 105-107] TRIGONOMETRIC FUNCTIONS 105 y = smx. Fig. 43. y = oosx. Fig. 44. y = sin X. Fig. 45. ^=cos a;. Fig. 46. 106 CALCULUS [CH. VII consists of an endless succession of exactly like bays alternately above and below the axis of as, Figs. 45, 46. Now if x is negative, — a; is positive. We put x' for - x. Then as x diminishes (algebraically) from zero, x' increases above zero. Now the graph shows that the values of sin x and sin x' are always equal in mag- nitude but opposite in sign. Hence the definition gives the equation sin a;' = — sin x, or sin ( — x) = — sin x. Again the graph of the cosine shows that the values of cos a; and cos x' are always equal and have the same sign, and the definition therefore gives the equation cos x' = cos X, or cos (-£») = cos X. 108. The other trigonometric functions are defined by means of the sine and cosine. We write down the defining equations sin a; ^ cos a; 1 1 ,,. tanx = , cota!=-: , seca! = , coseca;=^ — ...(1). cos X sin x cos x sin x ^ With a view to drawing the graph of tan x we observe that, 1 3 when x = ^ir, sin x = l, and, when x = s ""j sin x = -l. Generally sin X is either 1 or - 1 whenever x is an odd multiple of ^ ir. These are the greatest and least values of sin x. In like manner cos a; = 1 when a; = or x is an even multiple of tt, and cos a; = — 1 when X is an odd multiple of w. These are the greatest and least values of cos x. Whenever sin a; is 1 or — 1, cosa;= 0, and there- fore tana; is not defined by the above equation when x is an odd multiple of ^ x, for we cannot divide 1 or - 1 by 0. When x is nearly equal to ^ tt, sin x is nearly equal to 1 and cos a? is a very small number, positive if £c is a little less than ^ tt, negative if x is a little greater than ^ tt. Hence tan a; is a very great positive 107, 108] TRIGONOMETRIC FUNCTIONS 107 number when as is a little less than ^ ir, and — tan a; is a very great positive number when a; is a little greater than - tt. By the results of §§ 102, 107 we have tan (^7r + xj= — , tan (tt + a;) = tan r, tan(— a;)=— tana;...(2), and we therefore know the value of tan x for any value of x except odd multiples of ^ ir if we know the values of tan x between and g'jr (0 included, ^tt excluded). Just as in § 105 we may form a table of the values of tan x that correspond to values of x between and^ ir. In the table given on p. 108 the values of A tan X are correct to 3 places of decimals. The graph of tan x for values of x between and 1 '2 is shown in Fig. 47. For values of x between and - :5 T the graph is obtained by applying the rule tan ( — sc) = — tan x. The more complete graph (Fig. 48) is obtained by repeating the same curve in those strips of the plane which are bounded by pairs of consecutive lines given by equations of the form £c = an odd multiple of ^ ir, A in the same way as the complete graph of sin a; is obtained by repeat- ing the same curve in every strip of the plane that is bounded by a pair of consecutive lines along which x is a multiple of I-k. j/=tanx. Fig. 47. y. _j 7 t ■ 7 t 1 f 1 \ 1 I L I -A. T / 7 / /Jl X 108 CALCULUS [CH. VII Hi 0-1 0-2 0-3 0-4 0-5 0-6 0-7 tan X 0-100 0-203 0-309 0-423 0-546 0-684 0-842 X 0-8 0-9 1 1-1 1-2 1-3 1-4 1-5 tan X 1-030 1-260 1-557 1-965 2-572 3-602 5-798 14-101 V \ / i 7o / y = tan x. Fig. 48. 108, 109] TRIGONOMETRIC FUNCTIONS 109 Examples. 1. How do the graphs of sin ma; and ooamx differ from those of sin a; and cos x (i) when m > 1, (ii) when m < 1 ? 2. How do the graphs of a sin a; and a cos a; differ from those of sin a; and cos x (i) when a>l, (ii) when a< 1 ? 3. Draw roughly the graphs of 5 cos 2x and 2 sin - x. 4. Draw the graph of sin X- COS a;. 5. For what values of x is tana: equal to 1? for what values is it equal to -1? 109. "With a view to the differentiation of sin 6 and cos 6 we shall think of the sine and cosine of an angle of 6 radians, and, in the first instance, we shall take 6 to lie between and -= ir. Z The method which will be explained here turns upon two observations. (1) If P, Q are two points very near together on a circle (Fig- 49), x units of length the distance PQ, and I units of length the length of the arc PQ, then X is very nearly equal to I, and, when Q approaches P, so that x tends to zero, - tends to Fig. 49. A I as a limit. This result is really involved in the radian measure of angles. A formal proof will be found in Appendix VI. (2) In Fig. 50 let P, Q be two points on a circle, O the centre of the circle. Draifv the axes of x and y through O, draw the ordinates PN, QM, and the straight line PR parallel to the axis of X to meet QM in R. Produce this line to the right, as shown, draw the tangent PT, as shown, and let the axis of x cut the circle in A, as shown. Let the angles asOP, aj'PQ, k'PT be 6, |8, <^ radians. Let the lengths of the arcs AP and AQ be s and s + As units of length, let the coordinates of P be a, y and those 110 CALCULUS [CH. vir of Q be 03 + Ax, y + Ay, and let the length of the chord PQ be x units of length. Then Aa3 _ Av . „ — = cos B, — ^ = sin a, X X for Aa; and Ay have the right signs as well as the right mag- nitudes. Pig. 50. 110. We can now differentiate sin $ and cos 9. We have x = r cos 0, y = r sin 6, s = rO, where r is independent of 6, and therefore dx _d (cos ff) dy _d (sin 6) ds de de Now -^ and -— are the limits to which -r— and — ^ tend when as ds As As As tends to zero. But Ax Ax V „ V As X As ^ As As X ^* As 109-112] TRIGONOMETRIC FUNCTIONS 111 and the factors on the right all tend to known limits, for the limit of B is <^, and the limit of -^ is 1. Hence As dx , dy . , ^ = cos, :£=sin<^. 1 Further (jy = ■= ir + 0, so that cos <^ -• — sin 6, sin or < 0. 4 When the resistance is not very great, so that -jk^ < w', we put 4 Then ^^+™=(:,,^«) = 0, and we know that xe —a cos (mf — j8), where u, and /3 are constants depending on the initial displace- ment and initial velocity. Hence x is given in terms of i by an equation of the form x = ae cos (nj< — y8). The motion expressed by this equation is oscillatory. The body moves alternately from right to left and from left to right in a period of — seconds, but the extreme values of the displacement 120 CALCULUS [CH. VII continually diminish owing to the presence of the exponential factor e . The body settles down, as it were, into its position of equilibrium, by oscillating about it with excursions of con- tinually diminishing amount. Most natural oscillations take place in such a way as this. A motion of this kind would be described as a "damped harmonic motion." 119. When the resistance is very great the motion is quite different. If J i;2 > n2, we put ifc2-n2=m2. 4 Then ' — ' -n^\xe 1=0. Now we know (Ex. 8, p. 92) that this equation can be satisfied by putting where A and B are constants, and it can be proved (see § 127 below) that the equation cannot be satisfied in any other way. Hence we have a;=e"***(^e»n' + Be-'»'). We consider the case where the body is at first displaced a feet to the right, held for a moment, and then let go. Then we have x=a and ^- = when J=0. at Now if a;=c~**V«^ + ^«~'^). we find a=A+B, 0=A ( - ^k + mj+B (- ^k-mj, sothat ^ = i«(i+l)' ^=-g«(^-l)' Since m2<-rA;2 and m2=-J;2-n2, y 42- m2=n2 and 7ri>m, or r— >1. We 4 4 4 2 ' 2m see that x is always positive, a being positive. Again we find dx 1 di~ "2' il-^){l-^yi^---^' 118-120] TEIGONOMETRIC FUNCTIONS 121 and this formula shows that -p is always negative. The body therefore at always moves towards the position of equilibrium, but with continually diminishing velocity. It creeps towards this position, but never quite reaches it. 120. A special case of the problem concerning electric currents, considered in § 95, arises when the impressed electro- motive force is periodic. We shall suppose that, in the notation of § 95, E - Acos pt, where A and p are constants, and that di i = when t = 0. Then we have the equation L -5- + Ri = E, or di ,. A —. +f)i = - cos pt. dt L ^ This is the same as 'iiie'") A j^ — ^ — '- = - e™ cos p(, dt L ^' and (Ex. 8, p. 114) this gives L p^ + b^ To make i = when < = we must have a6 and therefore ,-M _ A / ^it P sin pt + b COS pt h \ '* "L^ p-' + b-' ~p' + bV' or i = -jj^^ {psmpt + b cos pt- be-"'). On substituting — for b, we find This formula gives the intensity of the alternating current produced in a circuit by a periodic electromotive force, the circuit being closed at the instant specified by < = 0, when the electro- motive force is a maximum. 122 CALCULUS [CH. VII Examples 1. A point P moveB on a circle (centre C) in such a way that the straight line CP turns with a uniform angular velocity, and from P a perpendicular PN is let fall upon a fixed diameter. Prove that N executes a simple har- monic motion. 2. Draw the displacement-time graph and the velocity-time graph for a point which executes a simple harmonic motion. 3. A body executes a simple harmonic motion according to the equation Initially it is displaced so that x=a, and it is let go from rest in this position. Express x in terms of t. Initially the body is undisplaced and it is projected so that -j:=u. Express x in terms of t. d^x (ix dv 4. If -^-^ + 'n?x=0, we may put -j;=", and then -j-= -n^x. Provethat dt fit CLt v^r-+n^x -,- — 0, and thence that ( -r- I +rfix^=oonsi. dt dt \ "' / Prove also that, if a body moves according to the equation (-j-]+ nV = const. it executes a simple harmonic motion. d^x 5. Prove that if two functions satisfy the equation -^+n^x=0, and yield the same values for x and -j- when t=0, their difference z satisfies the equation -^ + n^z = and yields zero values for z and j- when t=0. (dz\^ -J- 1 -Hi222=0, and thence that 2=0 for all positive values of t. Hence show that there is only one function which satisfies the conditions laid down for x. e. A body executes a damped harmonic motion according to the equa- 120] TRIGONOMETRIC FUNCTIONS 123 — ikt dx tion x = ae cos(mt-ji3). Prove that if x = a and ^=0 when t=0, »=a^(: 1 +2—2), and tan;8=^. Prove that the maxima of the dis- placement (sign disregarded) occur when t=0 or t=a multiple of — , and that each maximum is less than the preceding in the ratio e "* : 1 (sign disregarded). Prove further that the body passes through its equilibrium position at a series of instants separated by intervals of - seconds (a second being the unit of time) , and determine the first of these instants. Prove that, if any one of these instants is specified by the equation f =T, the velocity at that instant is mae' feet per second (a foot being the unit of length). 7. Draw the displacement-time graph for the body whose motion is described in Ex. 6. [Draw the locus of the maxima a; = ae and x= -ae , the graph is like a curve of sines with steadily diminishing maximum ordinates whose extremities lie alternately on these two curves.] 8. A gate supported on hinges swings to and fro, starting from rest in an extreme position in which its angular displacement from its equilibrium position is 45°, and next coming to rest in an extreme position in which its angular displacement is 40° on the other side of the equilibrium position, the time between these two positions of rest being 1 second. Assuming that its angular displacement 8 obeys the law of damped harmonic motion -ikt 1 [9 = ae cos {mt - 18)], determine the constants a, /3, ^ ft, m. [Results, 5 S; = 0-1178, m=7r, (3= 0-0375, a = 0-7909.] Find the angular velocity with which the gate first passes through its position of equilibrium, and the ratio in which this angular velocity is reduced at each subsequent passage. [Results : angular velocity = 2-34 radians Q per second approximately. Reduction - . ] 9. A point, which executes a damped harmonic motion, is observed to come to instantaneous rest in three successive positions, which are re- spectively 10 inches to the right of a point A, 9 inches to the left, 8 inches to the right. Find its equilibrium position. [Result ^ of an inch to the left of A.] 124 CALCULUS [CH. VII Rt lO. In the problem of § 120 after a time the term Re l in the expres- sion for i becomes very small and may be omitted. Prove that, when this is done, the maximum value of i is ,,„„ ^r-a, i and that the maxima of current follow the maxima of electromotive force after intervals of s seconds, where s is the smallest positive number which satisfies the equa- tion tan p« = 3" . CHAPTER VIII METHODS OF INTEGRATION It is very important to be able to integrate such simple functions as can be integrated easily. A number of methods which are applicable to various classes of functions will be exemplified. The method of resolution into partial fractions. 121. We begin with three examples, (1) /r^4q, we can put Vii-'-'h- and then x'+px + q=(x + ^p\ —fi = (^ + |^) + T}{(^ + i2')-v[, and if "=2^' + ')'. ^=-^V—1^ we have a;2+pa; + j = {a; + a) (x + V). 123-125] METHODS OF INTEGRATION 129 We may proceed by a method which is slightly different from that used in § 122, by putting x + 2P=». az + B-^ap m, , a« + /3 ^ 2 '^ Then we have -= ^—- = r 3 — , x^+px + q z' -y' and I -=- ^^dx=ia ]-, ^dz+ip --^ap] \-^ ^dz, J x^ +px + q j z^-'f \ % j ] z^-Y and ' iy""^ z + y' also 2y = a — b, z-y=x + b, z + y=x + a, , [ ax + ^ , 1 , , !. . ^ ~2°'^, x + b hence -= ^ — dx=- a.logi,(x^+px + q)+ r— log, . Jx^+px + ri 2 "'^ ^ ^' a-b ^ x + a The case where p^ <; 4gr will be considered in § 133. Examples Integrate the following (1) — (10) : — (^) ^234' '^' ^2^' *^' 4^231' (*) 552^' (a; + l)(a;-2)' ' ■' (l-.i) (2x-l) ' ''''x(x-\)' ^ ' x(x-l) {x-2) ' 3a + 4 ,,„, Sx + i (9) i^^iro. (10) a;2 + 3a;+2' ^ ' x'^ + Sx + 1' A Logarithmic Formula 125. The result found in Ex. 2, p. 92 gives us an important integral, viz. : ^ ■dx = \og,{x+^{x' + C)} (G). /. This is a standard form. It is valid whether C is positive or negative. L. c. 9 130 CALCULUS [CH. VIII 126. We can deduce the value of the integral We may put and x+i^p=z, but z^ + q — 2?",= a? + px + q, therefore 127. By means of the standard form (G) we may express y in terms of t when y satisfies the equation -j^ = 'n^!/. where m is constant. roi ""Tf SO that dw „ and dw „ du dt or Then «)2 -irfiy'^ is a constant and we may put wi-n^'^=m^C, where C is a constant. Thus we have (iy=™M?/^+c). 126, 127] METHODS OF INTEGRATION 131 Hence either m ^- = -p-z — =7 or m ^r- = - • dy-J(y^ + C) "' -dy- V(j/2 + C)- First suppose that m -=- = % ^/{y^ + C)^ Then mt = loge {t/ + ;^(!/2 + C)} +a, ■where a is a constant, er we have V(!/2 + C)+2/=e'"<-<'. Hence ,, „ ., = e"-™'. ■J{y^ + C)+y Multiply the numerator and denominator of the left-hand member by ^(3/2 + C)-y and note that W{y' + C]+y} W{y^ + c)-y} = c, we find s/'y'+<^)-y=,.-mt^ c or J{y^ + C)-y = Ce'^-™'. But J{f + C)+y = er"i-a. Therefore 2j/ = c™'-" - Cc"-™'. Put ^- - 1- Then If we supposed that m 3- = ,, .^ „, we could obtain the re&ult by dy ^/(r + C) writing - t for i. The form of the result is unaltered. We see that every dhi solution of the equation -Tj = vfiy must be of the form j/ = Ac'"* + Be-"", where A and B may be any constants. Examples Integrate the following (1) — (4): — 1111 ^{x2 + 2x)' ^'^{x^ + x)' ^ ' ^(2x2 + 3)' ''^{3x^-1)- 9—2 132 CALCULUS [CH. VIII Integration by Parts 128. An important method of integration is founded upon the rule for differentiating a product. We have d {uv) du dv dx dx dx ' which is the same as du uv /du , f dv , v-T- dx + lu-j- dx. We have therefore the equation f du , f dv ^ \v^!- dx=uv— lu-^dx. J dx J dx This formula is known as the rule of "integration by parts." 129. We consider some examples of integration by parts. (i) I loggxdx. Let u=x, «=logja:, then jloggXdx^xlogeX- j x-dx = X logj x-x. (ii) Ix^logeidx. As before it will be convenient to put «=logja!, because then -5- is a rational function, viz. -. The right form for u, is then given by the X du x^^^ requirement that a;" should be j-. We see that we ought to put u= — ^ . Then a;" loB,xdx = =■ log,* - / = - dx °' n+1 Jn + lx log^a;- This formula holds for all values of n except -1. For this case see Ex. 5 (6), p. 92. 128, 129] METHODS OF INTEGRATION 133 This example illustrates the choice of functions u, v in an integral which is being attacked by the method of integration by parts. Let | ^ (jc) i/r (x) dx be such an integral, and let ^ (x) be chosen as v, then i/f (x) = -=- , or u= jij/ (x) dx. If we are seeking such an integral as jf(x) dx, where /(x) can be resolved into two factors <^ (x), i/f (x) in many ways, we must choose a way in which one factor tj/ (x) is easy to integrate, and we generally try to choose a way in which the other factor <^ (x) shall be very much simplified by differentiation. (iii) I xe'^dx. Put u = e"^, v=x. Then jxe^dx = xe^- le'^dx=xe'^-e'^. (iv) / x'^e^dx. Put 11 = 6", v = x'^. Then /x»e":(icc = a»e"^-n|a;''-Vda;. (a) So I x'^^^dx=x'^M'' - (n - 1) I x'^-'^e'^dx. The process can be repeated until the required integral is found, n being a positive integer. A formula like (a) of this example, by which an integral is made to depend upon a simpler integral of a similar form, so that its value can be determined by repeating the same process a certain number of times, is called a "formula of reduction." W h{l + x^)dx. Put u=x, v=^{l + x^), then jj(i+x^)dx = x^{l + x^)~(jr^^dx. T> i 1=^ (l + xii)-! ,,, ,. 1 134 CALCULUS [CH. VIII Hence l^{l+x'i)dx=xj(l + x^) - |v/(l + a2) ''^ + f JTrTP) '*'"' and therefore 2 j s/(i. + x'^)dx=x ^{1 + x^) + I . ^. dx =x ^(l + x^) + loe,{x + ^(l + x^)}, or h(l+x^)dx = ^x^{l + x2) + ~log,{x + ^{l + x^)}. 130. We consider some examples of integrals involving trigonometric functions. (i) I a; cos a: dx. We integrate by parts, using the formula f dv , f du lu ^dx=uv- Iv-r- dx, J dx J dx and putting M=x, i; = sina;. We get ix coBxdx=x sin x- lsina;dx=a;sinii; + oosa;. (ii) jx'^coBxdx, Here we put i(=a;"', t) = sina; and find j x" cos xdx=x^ sinx- I nx^~^ sin adar. Again in I a;"^i sinrcda we put M=a;"-i, d = oo3 x and find - \x'"^^ avaxdx=x^~^aosx- I (n-l)a;''"2cosxda;. Hence vfe have la;"cosxdx=a:''sina:+«a;"~icosx-n (n-1) /x"-2co8xdx. This is a formula of reduction (cf. § 129 (iv)). (iii) \e'^ C03 l)xdx. Here we put = 6™, m=cos 6x, and find j ae'^ cos 6xdx = «°* cos bx + I 6c" sin 6xdx. 129-131] METHODS OF INTEGRATION 135 In like manner in | e."-^ sin 6a; we put « = «'", tt=sin 6x, and find I ac""^ sin bxdx=e^ sin 6a; - j 6 ive I e'^ aoabxdx=- e"'^ DOS bx +-2-jc'^sin6a;-6 I eo^eoa bxdxl ae"" sin bxdx=e'^ sin 6a; - | be'^'aoa bxdx. Hence we have e"" cos bxdx=- e'^ cos 6a; H — j e^ sin 6x, , , „, 6 sm bx + a cos 6* e<^oos6a;(j9!=e<" = — -= . a' + b' Examples 1. Integrate the following (1) — (11) : — (1) logiox, (2) a;21ogioa;, (3) ^-^ , (4) x^ e^ (5) a;^e^, (6) xc"^, (7) ccsinx, (8) s^oosa;, (9) a;^ sin x, (10) a;^ cos x, (11) a;3 sin a;. 2. Provethat /^(a;2 + A)da;=2[a;>y(a;2 + A) +Alog^{a;-HV(»^ + A)}]- 3. Find j^{x^+px + q)dx. 4. Obtain a formula of reduction for I e"'™a!''da;. 5. Obtain the value of I c"sin 6a;iia; by the method of § 130 (iii). Inverse Teigonometeic Functions 131. We can give a different form to some of the results in Ch. VII. Instead of writing y for sin 6 and x for 6 in the table of § 105, we may write x for sin 6 and y for 6, so that x= sin y. By means of the table we can draw the graph of y as a function of X for values of x lying between and 1 ; and by means of the relation sin (-«/) = — sin y we can continue the graph on the left- hand side of the axis of y, and so draw the graph of y as a function of x for values of x lying between — 1 and 0. Since the sine of a number always lies, between — 1 and 1 there 136 CALCULUS [CH. VIII is no value for y as a function of x except when x lies between -1 and 1. The extreme values of y are -jttt and ^ IT. The- equation a; = sin y by which this function is defined is written y = sin ~^ x. Fig. 54 is the graph of sin"' a;. ;?:; - - 2 - -. ■ 1 x..i:::: ;;:-:::-:::-;'■:: ::::;:::: .. <-. — ::::::::::::::::::: :::::::::;?!::::::::::::: :::: - --;■• ... . -/ tj- - ;-•- -- I 7 -■ -- — --,^-- ---- ^ :::::" ... .... i. . . .^ j. /=sin~'a;. Fig. 54. Fig. 55. In like manner by means of the equation x = tan y we may define a function of x. We write y= tan-^a;. The graph of the function tan"' x is shown in Pig. 55 for values of x lying between — 2'5 and 2-5. There is a definite value of y for every value of X. All the values of tan~^ x lie between — ^ ir and ^ ir. The functions sin-'x and tan~'a; are called the inverse sine and inverse tangent of x. We could, of course, define in the same way the inverse cosine, inverse secant and so on. 132. We can at once diflferentiate the inverse sine and inverse tangent. (i) Let x = siny, y=sm~'^x. 131-133] METHODS OF INTEGEATION 137 Then J=cosy=7(l-sm^2/)=V(l-a:^). Hence dy dx ij{\—a?)' Since the extreme values of y are - „ tt and -= it, cos y is a positive number for all the values of y that occur, and the square root must be taken to have the positive sign. (ii) Let X = tan y, y = tan ~^x. Then dx „ 1_ ^ . y" ( dy 1 dx . 1 cos" y + sin" y . sin" y , , „ -rr- = sec" y = — ;— = '^—z = 1 + — -? = 1 + tan" w = 1 + ; ay COS'' i/ cos^ y cos" y Hence j -i a- aa; 1 + a;" These results give us two new formulae of integration, viz. 1 /: ^/(l-«^) 1 dx = sin-^ X, h 1+oc" dx = tan~^ X. By using the rule for differentiating a function of a function (§ 25) we find the results /. c?a; = - tan~^ - (I). }a? + a? a a These are standard forms. 133. The chief purpose of the introduction of the inverse sine and inverse tangent into the elements of the Integral Calculus is the integration of such expressions as (as" - a;")'-i and (a" + a;") ~' and of others which can be reduced to them. 138 CALCULUS [CH. VIII We consider some examples of such reduction. By the method of § 124 we may write If q + jp' were negative the whole expression would be negative, and would not have a real square root. We therefore take q + jp^ to be positive, and put q + -p2-y2^ and y = /sj\q + l P^j- Then, writing x-^p = z, we have [ _j. , [ 1 ^ . _,z . , "-h I -II — ; -i\ "^= \—iT-!> — iT, dz=sin 1 -=8in 1 — —. s i . ]^[q^px-x-^) j^b'^-z') y /(q^\pi\ .... f ax + p , (u) / -5 '-- dx. J x'+px + q In § 124 we saw how to find this integral itp^>iq. We now take p2<4} and write and then, writing 1 +px + q=(x+-p\ +{n--^pA. isitive, we may put q-lP^=y^ and y=/^J\q-^pA, Since q ---ip^ is positive, we may put we have x + ^p=z, so that z^+y^=x^+px+q, J x^+px + - dx= I \ 5 —dz q J z^ + yS = » j^2 dz+[p-lap) J-^^ dz. Now j^2^2d'=2^°Se{^' + y')' I „ „ dz=-i J z' + y^ y and I -s 5 dz=- tan~i - 7 y 133, 134] METHODS OF INTEGBATION 139 On substituting a! + ^^for«, . /( g- jj)2 j for 7, a.ndx^+px + q{0T z^ + y'^, we find f ax + 3 1 /S-g"^ "'^2^ Examples 1. Prove that, when h is small, sin"! {x + h)'=Bin-^ x + -rrz: =7 , and that tau"i (x + h) = tan"i x + ;; ^ • ^ 1 + x^ 2. The sine of an angle is estimated as 0-4. Assuming that the error is not greater than O-OOl, find approximately the possible error of the angle in degrees, minutes and seconds. [Eesult 3' 45".] 3. The tangent of an angle is estimated as 1-5. Assuming that the error is not greater than O-OOl, find approximately the possible error of the angle in degrees, minutes and seconds. [Besult 1' 3"-5.] 4. Differentiate XBin~^x, -tan~ii. X 5. Integrate the following (1) — (7) : — (1) ^tA^' (2) -jj^^„ (3) ,„'_,, , (4) V(l-3^2)' ^ ' V(5-2x2)' W ^(i + x_^2y V' J(3 + 2x-x')' V(3 + a;-2a;2)' ^ ' x^ + s + l' ^' x'' + '2x + 3' 134. In many cases it is better to avoid introducing the inverse functions, and simply to use a substitution suggested by the previous results. For example, consider / -rj-^ j- dx. "We go back to first principles and observe that we are required to find a function y which shall satisfy the equation d2/_ 1 dx Jla' - X-) ' 140 CALCULUS [CH. VIII Now put a; = a sin 6, then J {a? -x^) = a cos 6, and -jz^'^ '"^^ ^> uu so that we have dy dydx 1 , , -T^=-T--m= a « COS e = 1, db dx dd a COS d and y = 6 satisfies the equation. That is to say, if we put x = as,m.d, all the values of | ,, „ t^ dx are included in the j J{a'-x') formula ^ + C, where C is an arbitrary constant. We might put a; = a cos <^ instead of a; = « sin 6. We should find in the same way -j^ = - 1, and so y = C' — , where C' is an arbitrary constant. The two forms can be reconciled by observing that if ^■r~ = $, then cos = sin 6. 135. We consider some examples of such substitutions, (i) jj[a^-x^)dx. Put x=asinfl, the integral becomes a^j eos^ BdB. We can find this integral from the result of Ex. 1 (7) on p. 114, or we may find it directly by integration by parts. We use the formula /^v _„ f du ,„ u-de=uv-jv^^de, putting «=cos 6, ti = 8in $, and getting I cos2 ede=isin 9 cor 9+ I sin^ede = sin ficos e+ I {1 - cos^ 0) dS = sinecose + 9- jcos^ede. Hence 2lcoB^ede = sine coee + 8, 134, 135] METHODS OF INTEGRATION 141 and j eos^ede = -{e + smecoiie). In like manner we should find / sin2 e de = -le-sine COS e). We find hence some important definite integrals, viz. Jo Jo * .'0 fa We may also find the definite integral / ^{a^ - x^ dx. The indefinite J integral is = -ja^sin"! - + x J (a^ - x^)\ , and, since this vanishes when x = 0, no constant need be added. As ^- increases from to a, sin~i - increases a from to ^ TT. Hence /: * (ii) \x^J[a?-x^)dx. Put x=asm6, we find \ x'^ ^(a? - x^)dx= a^lsvoi e oos^ ede, = aij{oos^e-eoaie)de. To find / cos* S tie we use the formula of integration by parts in the form written in Ex. (i) putting «=cos3fl, « = siu5, and getting, I cos*ede = cos3 61 sin fl + 1 3 cos2 e sin^ edd. = cos3 e sin e + 3 I (oos2 e - cos* 9) dd = cosS e sin e + 5 (e + sin 9 cos e) - 3 / cos* 6 de. Hence 4 I cos* 9d8 = cos^ 6 sin B +^(6 + sin 9 cos 9), 2' '2' r 1 3 and I COB* 9d9 = j cos^ ff sin 9 +5 (9 + sin 9 cos 9). 142 CALCULUS [CH. VIII We now find I a;2 J(a2 -x^)dx = a* f- (9 + sin 9 cos 9) - ij eos3 e sme+-(e + sin B cos 6)1 = a*f ^9 +-siu9 cose -J sinff oosSff ) . From the above work we deduce the important definite integral / COS* ede=^T. 16 We may also find the definite integral j x^^{a^-x^)dx. The in- r ■ J —a ± X 1 1ft definite integral is - a* sin-i - + -a^ xJ(a^-x^--;X (a^- x^f. We add a 8 a 8 i constant C and adjust C so that the integral may vanish when x= -a. X 1 1 When x= -a, sin-i - = - - ir, and therefore C = — ira*. As x increases from o, A lb X 11 - a to a, sin" 1 - increases from - „ ?r to ^ tt, and when x=a the value of the indefinite integral written above is j^ tra*. Hence /■ 16 x^^{a^-x^)dx=j^ira* + C=^TaK (iii) f-i^ de. ' J cos 8 We know that j -j—--^dx =log^{x + J [l + x'^)}. Now put x=tau9. We find J{l + x^) = sece, ^ = Bee^e, and therefore | -77, 5r dx= sea8d0=i —-5 de. We have therefore the result / de= loge (tan e + sec 9) , or, as it may be written, \. 1 ^^ , l + sin9 COS B cos 8 13 13 135] METHODS OF INTEGRATION 143 Examples 1. Integrate the foUowing (1)— (5) : — (1) sm2a;oosa;, (2) oos^ajsinx, (3) sin' a;, (4) oos^x, (5) sin* a;. 2. Prove that ( sm*xdx — ^Tr. I. Prove that / - — dx=loge — j sin a: 144 CALCULUS [CH. VIII Additional Exercisbs [It is not intended that these examples should all be worked before the rest of the boob is read. The student will find it profitable to work a few of these every day for the sake of practice.] 1. Integrate with respect to x the following (1) — (50) : x^ + ix + 1' ^ ' x-^ + ix + 2' ^' x2 + 4a; + 3' (10) ;3xix-.. (11) ;2v|rT«' (12) ^'''"^ x^ + ix + 5' ' ' x^ + 4x + 6' ^ ' x^ + Ax + T (13) ,, i^,-„\w„ .^ . (1*) (a;-l)(a;-2)(x-3)' ^ ' (a; - 1) (a; - 2) (3 - x) ' *^®' (a;-l)(2-a;)(3-ii;)' (^^' V(2x2-5)' '"' V(2^'-6)' *^^' n7(612^' ^"^ J(5-2x^)' (20) x/(2x2-3), (21) xV(2a;2 + 3), (22) J{2-3x^), (23) a;J(l-3a;2), (24) x^ ^{2x^ + 5), (25) a;V(3-2ar2), (26) (a;2 + 2a; + 3)V(l + ;t), (27) {x^-3x + 2) s/{l-x), (28) xV(2a; + a:2), (29) {x~l}J{2x-x% 1 X *^°' ^(2a:2 + 2x-l)' *^^' ;^(2a;2 - 2x - 1) ' (^^' ^(2 + 2.T-x2J' _(^^' ^(l + x-2x2)' (34) VCa^^ + aJ-l)' (35) :<^J(a;2+a; + l), (36) ^/(1 + a; - a;2), (37) xJil-x-x^), (38) sin-la: (39) a;sin-ix, (40) x^sin-ix, (41) x^gin-ix, (42) tan-ix, (43) xtau-ia:, (44) xHan-ix, (45) x^tan-'-x, (46) e^^cosSa;, (47) e=^ sin2 x, (48) c"'" cos^ x, (49) «"* sin^ x, (50) («"^ + c"'") sin X cos x. 135] METHODS OF INTEGRATION 145 2. Compute approximate values of the following definite integrals (1) — (10):- (7) 10) — !- is not zero or a multiple of ir. 137. The last result shows that the gradient of a curve at a point, or the gradient of the tangent to the curve at the point, is the (trigonometrical) tangent of the angle which the (geometrical) tangent to the curve at the point makes with the axis of x. If the (geometrical) tangent meets the axis of a; in a point, say T, the angle in question is supposed to be generated by a straight line drawn from T along the axis of x to the right, and revolving in the counterclockwise sense from this position into the position of the tangent. This statement is necessary in o^der to secure that the (trigono- metrical) tangent has the same sign as the gradient. 138. We apply the formula IV {^+(1)1 to find the length of the arc AP of a circle given by the equation x^+y^=r^, where P is a point (x, y) in the first quadrant as shown in Fig. S8. The sign has been taken to be +, and this is shown by the figure to be right. We have dx _ y_ dy' Hence di 1 + [dxy_ r' \dy)'~r^-y^ dy ^(r^-y'^y Fig. 58. and therefore '='\jW^ dy + C=rsiu-^-+C. Also, when y = 0, s=0, and therefore s=rsin""i- . r If the angle AOP is B radians, y is rsin 8, and our result is the same as the known formula s=tB. 139. As an additional example we find the length of an arc of a para- bola. Let the equation to the parabola hey=x^, and let us find the length of 136-140] RESULTS CONNECTED WITH ARCS OF CURVES 149 the arc contained between the origin and any point {x, y) on the curve. We may take s and x to increase together. We have y=x\ ^=2a!, ^=^{l + ix^), and s=0 when x=0. Hence s=\j(l + 4x2) dx + C =^x V(l +4x2) +liog^ {2x + V(l + 4a;2)} +C, where C is determined by putting s and x both equal to 0. We find C=0, and therefore 8=5x^(1 + 4x2) +2log, {2x + V(l+4x2)}. Curvature 140. Let AB be an arc of a curve, P a point on this arc. We can think of P as travelling along the curve from A to B. From a point O draw a straight line Op parallel to the tangent at P, drawn in the sense of description of the curve. We can take the arc AB to be so short that (i) Op turns about O always in the same sense, (ii) the angle through which it turns between its extreme positions Oa, Ob is acute (Fig. 59). Let the length of the arc AB be I units of length, and the angle aOh be y radians. The angle y radians is the angle through which the tangent turns 150 CALCULUS [CH. IX as P moves from A to B. It may properly be called the " total curvature " of the arc AB. The quotient y measures, in radians per unit of length, a quantity which may properly be called the " average curvature " of the arc AB. The limit k to which this quotient tends as B is brought nearer and nearer to A is defined to be the measure, in radians per unit of length, of the " curva- ture " of the curve at the point A. 141. If the curve is a circle (Fig. 60), the angle which the arc AB subtends at the centre is y radians, and, if the radius is r units of length, I = yr. Therefore k = - . Therefore the curvature of a circle of radius r units of length is the same at every point, and equal to - radians per unit of length. 142. In the case of any curve the length - units of length is the " radius of curvature " of the curve at the point A. We write p for — . K 140-144] RESULTS CONNECTED WITH ARCS OF CUEVES 151 On the normal at A take a point C so that (i) the part of the curve near to A is concave to C, (ii) the length of AC is p units of length. With centre C and radius p units of length describe a circle. The point C is called the " centre of curvature " of the curve at A, and the circle is called the " circle of curvature '' of the curve at A. 143. Let s and <^ be defined as in § 136. Then <^ may in- crease as s increases, or it may diminish. When s becomes s + As, and ^ becomes <^ + A<^, the absolute value of As (sign disregarded) is the number of units of length in the length of an arc, and the absolute value of A<^ is the number of radians in the total curvature of this arc. Hence the absolute value of -^ as 1 IS K or -. P 144. When the curve is given by an equation of the form y =f(x), we can find a formula for p. We have dy , , dx -f = tan d), ^=- = cos 0. dx ^ ds ^ „ d^y . d d d Hence ^, = sec^ is 1 1 + (-^ j j- , and therefore H^^C dy\Y^d^ dx)] da?' where the + sign is to be taken when -5-^ is positive, the — sign when -5-^ 18 negative. 152 CALCULUS [CH. IX 145. As an example consider the parabola given by the equation y=x^. We have 1 S and therefore p== (1 + ix^) . 148, If a curve differs very little from a straight line, we may take the straight line to be the axis of x. Then y and -p are very small at every point of the curve, and we have the approximate equation l._ d2v p "• * dx2 ' where the upper or lower sign is to be taken according as -j-^« is positive or negative. For example if the equation to the curve iiy=a sin nx, where a is a very small constant, - is approximately equal to the absolute value of arfi sin nx or nhj (sign disregarded). Area op a surface op revolution 147. We find the area of the surface of a cone. Let the cone be generated by a right-angled triangle ACB revolving round the side AC (Fig. 61). Let the side AB contain I units of length, and the side BC, r units of length, and let the area of the curved surface be S units of area. The base of the cone is a circle v^ith C for centre, and its radius is r units of length. In this circle we suppose inscribed a polygon of a large number of sides, and join the vertices P, Q, ... of the polygon to the vertex A of the cone. We thus form a pyramid on a polygonal base. Let the length of the perpendicular let fall from A upon any side PQ of the polygon be p units of length. If the length of PQ is s units of length, the area of the triangle APQ is ■= ps units of area. 145-147] RESULTS CONNECTED WITH ARCS OF CURVES 153 Now let the number of vertices of the polygon be increased, and the lengths of all the sides PQ be diminished indefinitely. The pyramid tends to coincidence with the cone. The sum of the lengths of all the sides of the polygon tends to a limit, which — -pQ Fig. 61. is 2irr units of length, the number p tends to a limit, which is I ; and therefore the sum of the areas of the triangular faces of the pyramid tends to a limit, which is -x {2irr) I units of area. Hence we have S = irrl. If a denotes the numerical measure of the angle BAC (so that 2a is the measure of the vertical angle) we have y=sina, and S = 7rZ2 sin o. Let B he displaced along a generating line of the cone to B' (Fig. 62), so that I becomes Z + AI, and r becomes r + Ar. Then a is unaltered, and S becomes S+AS, where AS = 27r( I +9 Ai j sin a . AJ, or AS = 27r(r +=A)-)A?. = '2.^r\r^■■^^r\L 154 [CH. IX Fig. 62. 148. We take next the question of the area of a belt of a sphere contained between two parallel planes. We think of the sphere as generated by a semicircle revolving about its base. We take the axis of x along the base of the semicircle as in O NM Fig. 63. Fig. 63. We take a point P on the semicircle, and draw the ordinate PN. Let x, y be the coordinates of P. As the semicircle revolves, the point P traces out a circle whose centre is N and whose radius is y units of length. Let the area of the portion 148] RESULTS CONNECTED WITH ARCS OF CURVES 155 of the sphere, which contains the point O and is cut off by this circle, be S units of area, and let the length of the arc OP be s units of length. When P moves to Q, so that s becomes s + As, S becomes S + AS. As the semicircle revolves, the arc PQ, of length As units of length, traces out a belt of the sphere, and its area is AS units of area. The chord PQ traces out a belt of a cone. Let the length of the chord PQ be x units of length, and let the area of the belt of the cone in question be a- units of area. We know that, as As tends to zero, — tends to a limit, which is 27r2/. Now — tends to 1 as a limit, and so also does — . Hence -r— tends X cr As to a limit, which is 2Try, or we have Let the radius of the sphere be a units of length. Then Fig. 64 shows that in every position of P duo , _ . y -=- = cos a; PT = sin PCN = - , as a and therefore ax Since S = when as = 0, we have S = 2irax. 156 CALCULUS [CH. IX It follows that the area of the belt of the sphere contained between two parallel planes, at a distance apart equal to d units of length, is 2irad units of area. The area of the whole sphere is ^wa^ units of area. 149. The reasoning by which we established the equation ds does not depend upon the revolving curve being a semicircle. It may be applied to the surface traced out by the revolution of any curve about the axis of x. Examples 1. The radius of the circular base of a cone is r units of length, and the vertical angle of the cone is 2a radians. Express the area of the curved surface in terms of r and a. 2. A piece of paper in the form of a quadrant of a circle is wrapped round a cone, so that the centre of the circle is at the vertex of the cone, and the two bounding semidiameters of the quadrant are seen to lie along the same generating line of the cone. Find the vertical angle of the cone. [Besult: 28° 57' approximately.] y 3. How much cloth is required to cover a lawn-tennis ball of diameter 2J inches? [Besult : 19-63 square inches ap- proximately.] 4. Let the parabola whose equation is y=^x revolve about the axis of x (Fig. 65) . Prove with the notation of §§ 148, 149 that —=Tj{l + ix). Hence show that if the length of ON is 2 units of length, the area of the curved surface traced out by the revolution of the arc OP is 13'6136 units of area approximately. CHAPTEE X THE DEFINITE INTEGRAL AS THE LIMIT OP A SUM 150. The object of such investigations as that in § 71 of the area under a curve is not so much the mensuration of plane figures as a graphic representation of integrals. The number of units of area z in the area bounded above by the graph of y=f(x), below by the axis of x, to the right by a moving ordinate (a;) and to the left by a fixed ordinate (a) is expressed as a function of x by the sum of the integral jf{x) dx and a constant C ; and this constant is determined by the condition that z = when x = a. The value of z when a3 = 6 is expressed by fb the definite integral / /(x) dx. We may think of the indefinite Ja integral as a definite integral with a variable upper limit, and we may represent it graphically by an area. In general the only way of finding either the definite integral or the area is to find the indefinite integral by the methods of the Integral Calculus. But if we could find the area by any other method, we should find the value of the definite integi-al without first finding the indefinite integral. If the indefinite integral is one that we cannot find, we may be able to calculate the area approximately, and then we shall know the integral approxi- mately. 158 CALCULUS [CH. X __ T .^ ■^ o > ^ 5 ^ Z- L 1 N M Fig. 66. 151. The most obvious way of approximating to the area is to draw the graph on squared paper and count the squares that are inside it. The area will be a little greater than the sum of the areas of the in- cluded squares. We can im- prove the approximation very much by quite simple means. (i) Instead of merely counting the squares complete the rectangles such as PNML (Fig. 66). The sum of the areas of these rectangles is a better approximation to the area than the sum of the areas of the included squares. Let the unit of length be such that the distance between consecutive ruled lines is h units of length. If the figure is drawn on a large scale A is a small fraction. Let y„ y,,...y^ be the ^-coordinates of the first n points. The corresponding sum of areas is ^(2/1 + 2/3+ ••■+yn-i) units of area. (ii) Instead of completing the rectangles such as PLMN complete the trapeziums such as PQMN. The area of the trapezium between the rth and (r-i-l)th ordinates is units of area, and the sum of the areas of the trapeziums gives a good approximation to the required area. The approximate expression for the number of units of area is 2 h {{y^ + ya) + (2/2 + S's) + • ■ • + {yn-2 + Vn-l) + (2/n-l + 2/n)} or K^{2/i + 2/» + 2(y2 + 2/3+-"+2/«-i)}- 151, 152] DEFINITE INTEGRAL AS LIMIT OF A SUM 159 (iii) Insteaxi of joining two such points as P, Q by a straight line, take the points in threes, P, Q, R, then R, S, T, and so on, and suppose each set of three to lie on a parabola whose axis is parallel to the axis of y (§ 35). This will give a still better approximation. Now we know (§ 73) that the area between the first and third ordinates, the axis of x and the corresponding parabola, is and a like formula holds for all the corresponding pieces of area. Let there be 2?i+ 1 points such as P, Q, ..., then the approximate formula for the number of units of area is 3 ^ {(2/1 + 2/3 + 4^2) + (2/3 + ys + 4^4) + ■ • ■ + {vm-i + ya+i + ^y»)], 3^{2/i+yai+i + 2(2/3 + 2/5+ •■■+2/2K-i) + 4(2/2 + 2/4+ ••• +y^)Y Inside the bracket we have (a) the sum of the first and last y's, (6) twice the sum of all the other y's with odd suffixes, (c) four times the sum of all the y'a with even suffixes. This rule for calculating areas is known as Simpson's Rule. 152. We may use Simpson's Eule to approximate to ir. Take a circle •whose radius is 1 unit of length, draw the axes of x and y as in Fig. 67, mark the points I - ^ , J and ( ^ , 01, which are C and D, draw the ordi- 160 CALCULUS [CH. X nates CE, DF. Then EF = CD=OA, and the triangle EOF is equilateral, so that the angle EOF ia^jr radians. Also the length of EC or DF is V (■'■~\2/['"^2'^* "°'*^ °^ length. The area of the curvilinear figure bounded by the arc EF, the ordiuates EC, FD and the axis of x is the sum of the areas of the sector OEF, and the triangles EOC, FOD, or it is ii..Q.lf) units of < To evaluate the same area by Simpson's Eule, divide CD into ten equal pieces. We have y=^{l-x'), xi=--05, 3:2= -0-4, ... a!n = 0-5. We find (Ex. 16, p. 15) for,gft{j/i + j^ii + 2(y3+2/5+3/7 + i/j) + 4(r/2 + j/4 + 3'o+2/8 + 2/io)} the value 0-95661 to five decimal places, and we find -7 \/3= 0-43301 to five decimal places. Hence 1 and 5 7r= 0-52360 to five decimal places, 7r= 3-1416 to four decimal places. 1S3. In like manner we may use Simpson's Bule to approximate to logiofi. We consider the area contained between the hyperbola whose equation is 2/ = - (Fig. 68), the ordinates whose equations are x=l and x=2, and the axis of X. We know that this is log^ 2 unite of area. We divide the portion of the axis of x contained between the two extreme ordinates into ten equal pieces,sothatj/i = l,j/2= jTj, ...2/11=^ , /i=0-l and calculate Fig. 68. ^h[yi + yn + 2{y3 + ys + y,+ya) + i{y2 + yi + ye+yi + yio)]- We find (Ex. 17, p. 15) the result 0-69315. Now logio 2 = 0-30103 and 30103 logioC=c7r5iK= 0-43429 to 5 places. This result gives « = 2"718B to 4 places. 154. We return to the method of approximating to the area by summing the areas of rectangles, and observe that the 153, 154] DEFINITE INTEGRAL AS LIMIT OF A SUM 161 approximation may be improved almost as much as may be wished by taking more numerous intermediate ordinates at shorter distances apart. The development of this remark leads to an important result. To fix ideas we take fix) to be positive and to increase as X increases. Let A and B be the points of the graph of y =f{x) R Q N M Fig. 69. at which x = a and x = b, and let AC, BD be the ordinates of these points (Fig. 69). Between C and D take a number of inter- mediate points such as N, M, draw the ordinates such as NP, MQ, and complete the rectangles PNMS, QMNR. The area contained L. c. 11 162 CALCULUS [CH. X between the arc PQ, the ordinates PN, QM, and the axis of x is intermediate between the areas of these two rectangles. The sum of the areas of all the "exterior " rectangles, such as QMNR, exceeds the sum of the areas of all the "interior" rectangles, such as PNMS, by the area of a rectangle whose height is the difference of the extreme ordinates BD, AC and whose base is less than the greatest of the segments such as NM. As the number of the intermediate points such as N is increased, and the lengths of all the segments such as NM are diminished, the difference between the sum of the areas of all the exterior rectangles and the sum of the areas of all the interior rectangles continually diminishes, and can be made as small as we please. The common limit of both these sums of areas is the area of the curvilinear figure ACDB. Now this result means that if we write x^ for a and x„ for b, and take w — 1 intermediate values asj, ica, ... x„_i, the sum f(Xo){Xi-Xf,)+/{Xi){Xi-X^)+ ...+f{x^_-,){Xi,-X„.j)+... +/K-i)(a'i.-'B»i-i) is an approximation to the definite integral I /(x) dx, and so also J a is the sum /(aji) (!Bi - aj„) +f(x^) {x^ - Xi) + . . . +f{xj,) {x„ - x^.,) + ■■■ +/(«,!) (»»-aJn-i); and, further, that if we invent any rule for increasing the number M — 1 of the intermediate values and spacing them out, so that a-j shall always be a and a;„ shall always be b, and all the differences aij, - a3j;_i shall tend to zero, both these sums tend to the fb same limit, and this limit is | /{x) dx. Further, if xV denotes Ja a value of x lying between a;^_i and a;,., the sum /(a^') (Xi - a;„) +/«) {x^ - Xi) + • ■ • +/«) {^ - a;*-i) + ... +/«)(a!„-a;„_i) 154-156] DEFINITE INTEGRAL AS LIMIT OF A SUM 163 rb tends to the same limit, viz. : I / («) dx. Such a sum may be Ja written a where the symbol 2 (sigma) means " the sum of such terms as," the letters written above and below indicate the first and last values of x, and Ax denotes in a general way the difference between two consecutive values of a; in a series of values placed between a and b. 155. The result would not be altered if f(x) were to diminish as x increases, or if it were to increase in some parts of the range indicated by a, b and diminish in other parts. Nor is it necessary that/(:E) should always be positive. The summations for the ranges in which it is positive and those in which it is negative could be performed separately. Such limits of sums present themselves naturally not only in problems concerned with areas but also in many problems that have nothing to do with areas. The result that such limits can be evaluated as definite integrals enables us to find them whenever we can find the corresponding indefinite integrals. 156. In the case of the area of a figure {§ 75) we may draw across the figure a number of straight lines parallel to the axis of y. If Y units of length is the length intercepted by the figure upon one of these lines, x the j;-eoordinate of any point on that line. Ax units of length the breadth of a strip between two consecutive lines of the set, the area of the strip of the figure between the two lines can be replaced by the area of a rectangle approximately equal to YAx units of area. The exact number is expressible as Y'Aa;, where Y' tends to Y as a limit when Ax tends to zero. The limit of the sum SYAx, where the summation refers to all the strips is the number of units of area in the area of the figure. This result is equivalent to the one found in § 75. In the case of the volume of a solid (§ 79) we may think of the solid as divided into a large number of slabs or laminae by planes drawn parallel to each other. If Z units of area is the area of the section of the solid by one 11—2 164 CALCULUS [oh. X of the planes, Ax units of length the distance of this plane from the next, the volume of the lamina is approximately equal to ZAs units of volume. The exact number is expressible as Z'Ax, where Z' tends to Z as a limit when Ax tends to zero. The limit of the sum SZAx, where the summation refers to all the laminae, is the number of units of volume of the solid. This result is equivalent to that found in § 79. In the case of the length of an arc AB of a curve we may begin by mark- ing on the curve a number n - 1 of points Pj , P^, ... P„_i between the two points A and B. The length, Xr units of length, of the chord P,._i P,. is given by the equation Xr' = (x,. - x,_i)2 + {y^ - 2/,_i)2, where x,., yy&ie the coordinates of P,. This is equivalent to the general formula x2=(Ax)2 + (Ai/)2 of § 136. The number of units of length in the length AB is the limit to which Sx tends when the number n is increased indefinitely, and the lengths of all the chords are diminished indefinitely. In case x always increases, as a point travels along the curve from A to B, we have the result ■'i>m\''- s being the number of units of length in the length of AB, and a, b the x-co- ordinates of A, B. The result may be expressed by saying that the length of a curve is the limit of the length of the perimeter of an inscribed polygon. The limit is arrived at by making the lengths of all the sides of the polygon tend to zero. In the ease of a surface of revolution, generated by the revolution of an arc AB about an axis, we divide AB as before. The chord joining two of the points such as P,_i and P,. generates a belt of a cone. Let the area of this belt be o-^ units of area. Then the area of the surface of revolution is the limit to which Sir, tends. If the curve revolves about the axis of x, ff^ is approximately equal to 27n/,x,. The exact value is expressible as 27ry/xr where y^' tends to y^ as a limit when Xr tends to zero. This leads to the result obtained in §§ 148, 149. The result may be otherwise expressed by saying that the area of the surface is the limit of the area of a different surface, viz. : one obtained by inscribing a polygon in the generating curve and making this polygon revolve about the axis. The limit is arrived at in the same way as in the case of an arc. In the next Chapter we shall consider further examples of limits of sums evaluated as definite integrals. CHAPTEE XI SOME APPLICATIONS OP DEFINITE INTEGRALS TO MECHANICS 157. We may use the Integral Calculus to determine the position of the centre of gravity of a body. For this purpose we begin by recalling the meaning attached in Statics to certain ternas. Fig. 70. We think of a force of magnitude W lbs. ^ acting downwards in a vertical straight line NM, and we think of a horizontal ^ In regard to the units of force &c., see Appendix VII. 166 CALCULUS [CH. XI straight line AB which does not intersect N M. In Fig. 70 AB is to be thought of as being at right angles to the plane of the paper. A horizontal plane passing through AB would meet NM in a point, say N, and a perpendicular from N to AB would meet AB in a point, say A. If the length of AN is x feet, the product kW is the measure in Ib.-ft. units of the " moment " of the force about the straight line AB. We could make pass through AB and NM two vertical planes parallel to each other. The x which occurs in the product ojW is the number of feet in the distance between these two planes. It is usual to give a sign to the moment so as to indicate the sense in which the force tends to turn a body about the straight line AB. We could take an origin O on the vertical plane through AB, and an axis of x at right angles to this plane. The moment with its proper sign is still measured in Ib.-ft units by the product aW, if x means the ^-coordinate of the point L in which the vertical plane, passing through NM and parallel to AB, is cut by the axis of x. 158. The force of the Earth's gravity acting on a body does not act at one point of the body more than at another ; but, if the body is rigid, it can be supported by a single force, without undergoing any change of size or shape, provided this force is directed vertically upwards and acts in the proper line. How- ever the body may be turned about, there is one point (fixed with respect to the body) through which the line in question always passes. This point is the " centre of gravity '' of the body. For many purposes we may regard the force of the Earth's gravity acting on a body as a single force acting down- wards in the vertical straight line which passes through the centre of gravity. This force is often called the " weight " of the body. 159. We may think of a body as consisting of several parts. Each part is a body having a certain weight and a certain centre 157-160] APPLICATIONS TO MECHANICS 167 of gravity. The rule by which the centre of gravity of a body is determined is this : — The moment (about any horizontal axis) of the weight of the body, acting at the centre of gravity of the body, is equal to the sum of the moments (about the same axis) of the weights of the parts, acting at the centres of gravity of the parts. Let the .r-coordinates of the centres of gravity of the parts be a\', scj', ..., and their weights Wj, Wj, ... lbs. Let the a^coordinate of the centre of gravity of the body be Xq. Then we have the equation Xq (Wj + Wj + ...) =aJi'Wi + X2'V/^+ ..., or, as it may be written, x %\N = %x'VJ. The coefficient 2W in the left-hand member of this equation is the numerical measure in lbs. of the weight of the body. In applying this formula to find the centre of gravity of a specified body we shall think of the body as "homogeneous " or " uniform," meaning that the weight of any volume is pro- portional to the volume. We evaluate such a sum as ^x'VJ by passing to a limit, the volumes of all the parts being diminished indefinitely. The limit of such a sum is expressed by a definite integral. 160. As an example we find the centre of gravity of a hemisphere. Let the hemisphere be placed with its plane base vertical, and let the radius be r feet. The section of the surface by a plane parallel to the base, and at a distance X feet from it, is a circle, and the area of the circle is v (r' — x') square feet (Fig. 71). The volume of the slice between two such planes specified by x and x + Ax is intermediate between tt (r* — x?) Ax and IT {r^ -{x + Axf] Ax ^^S- 71. cubic feet. The distance, k' feet, of the centre of gravity of the 168 CALCULUS [on. xi slice from the base is intermediate between x and x + Aa; feet. Let the weight of a cubic foot of the substance be o- lbs. As all the numbers Ace tend to zero, the sum expressed by Sa-'W tends to a limit, which is /. T crir {r^ — ay') xdx, 2 and 2W is -^ irr^o-. Hence we have o 2 / r^ r*\ 1 and '"g=8''" 3 The distance of the centre of gravity from the base is ^ of the radius. The centre of gravity obviously lies on the straight line drawn through the centre of the base at right angles to the base. 161, In the case of a body in general, if a, h are the least and greatest values of x that occur, if the area of the section of the body by a plane at a distance x feet from the origin is Z square feet, and the volume of the body V cubic feet, the same reasoning leads to the result b ■I J a Xq\/= I Zxdx. 162. For another example, let the body be a pyramid on a triangolar base. Let the area of the base be B square feet, and the height p feet. As in % 80 the area of the section specified by x is B -^ square feet, and the volume of the pyramid is 5 Bj) cubic feet. We have 1 fP x^ 1 and ^ ~i^' 160-163] APPLICATIONS TO MECHANICS 169 Hence the distance of the centre of gravity from the base is j of the distance of the opposite vertex from the base. This result holds whichever face we take as base. It is equivalent to the result that the centre of gravity of the pyramid coincides with that of four equal weights placed at its 163. The point v^hich we determine as the centre of gravity of a uniform solid body is a sort of centre of a solid figure, the figure of the body. It is called the " centroid " of the figure. In like manner a plane figure has a centroid, which coincides with the centre of gravity of a very thin slab or lamina, having the shape of the figure, and having a uniform small thickness as well as a uniform density. Let the area of the figure be S units of area, and let the least and greatest values of x that occur in the figure be Xq and x^. Let any straight line be drawn across the figure parallel to the axis of y, and let Y units of length be the length intercepted by the figure upon this straight line, so that Y is a function of x (Fig. 72). Then the x of the centroid is given by the equation xWdx. Fig. 72. 170 CALCULUS [CH. XI In like manner, if y, and y^ are the least and greatest values of y that occur in the figure, and X units of length is the length cut out by the figure on a straight line parallel to the axis of x, so that X is a function of y (Fig. 73), the y of the centroid is given by the equation yaS = y^ydy. Fig. 73. Examples 1. Use the method of the Integral Calculus to prove that the distance of the centroid of a triangle from any side of the triangle is - of the distance of the opposite vertex from that side, 2. In the case of a semicircle, let the axis of y be the bounding diameter and the axis of x the straight line drawn at right angles to it through the centre of the circle, r units of length the radius. In the notation of § 163 !/q=0 by symmetry, and S = -7rr2, ^ = ij{r^-x^), X(,=(i, Xi — r. Prove that 4 a:- = r— r, or the distance of the centroid from the centre is 0-4244 of the OTT radius approximately. 3. In the case of a segment of a circle, let the bounding chord of the seg- ment subtend at the centre an angle 2a radians, let the origin be at the centre of the circle, and let the axis of x bisect the bounding chord. Prove that « o / • , r. 2 sin3 a S=r2(o-smocoso), Vo = Oi ^r=s'' = • V /> »G ' G 8 o - sm a cos a 163, 164J APPLICATIONS TO MECHANICS 171 4. In the case of a sector of a circle, the two bounding semi-diameters containing an angle 2a radians, let the origin be at the centre of the circle, and let the axis of x bisect the angle of the sector. Prove that ^0=0 and 3 sin a 5. We may define the eentroid {Xq, y^) of an arc by the formulae Xq.I= I xds, yQ.l= I yds, where the length of the arc is I units of i So J So length, X, y are the coordinates of a point of the arc distant s units of length along the curve from some fixed point of the curve, and Sq , si are the extreme values of s. The eentroid of the arc coincides with the centre of gravity of a piece of uniform wire bent into the shape of the arc. Prove that in the case of a circular arc, subtending an angle 2a radians at the centre of the circle, the eentroid is on the straight line drawn through the centre to bisect this angle, and is at a distance from the centre equal to of the radius. Centres of Pressure 164. Centroids are important in Hydrostatics as well as in Statics. The pressure of a fluid at a point is measured as so many lbs. per square foot. The pressure of water (at rest) at a point distant y feet below the surface of the water is wy lbs. per square foot, where w denotes the weight of a cubic foot of water. A unit of pressure is one lb. per square foot. Consider the force with which water presses against a limited portion of a vertical plane with which it is in contact. This force is called the " resultant thrust " of the water against the portion of the plane. We shall refer to the portion of the plane as the "figure." Let the area of the figure be S units of area. The plane cuts the water surface in a horizontal line. On this line take an origin O, and draw the axis of y vertically down- wards. The figure will be contained between two horizontal lines at depths a and b feet below the water line (Fig. 74). Any intermediate horizontal line will cut the boundary of the figure, and there will be cut out upon it a certain length, say X feet. The pressure at any point between the lines specified by y and 172 CALCULUS [CH. XI y + Ay will be intermediate between vjy and iv(y + A.y) lbs. per square foot, and we may take it to be toy' lbs. per square foot, where y' is some number between y and y + Ay. The area be- tween the two lines is X'Ay square feet, where X' lies between the greatest and least values of X that belong to lines between those specified by y and y + Ay. Hence the resultant thrust on this part of the figure is ivy'X'A.yVos. The resultant thrust on the whole figure is, in lbs., b w^y'X'Ay. Fig. 74. As all the numbers Ay tend to zero this passes over into w I yXdy. Ja But if 3/_ is the y of the centroid of the figure we have, as in § 163, y S = I yy^dy. Ja 164, 165] / APPLICATIONS TO MECHANICS 173 The resultant thrust is therefore wy^S lbs. 'Now the pressure at the centroid is wi/q lbs. per square foot. Hence the resultant thrust is the same as it would be if the pressure were the same at all points of the figure and equal to the pressure at the centroid. 165. The thrust of the water against the figure is statically equivalent to a single force. We have just found the magnitude of this force. Its line of action cuts the plane of the figure in a point, called the "centre of pressure." The depth of the centre of pressure below the water line is determined by the rule : — If the figure is regarded as made up of parts, the moment (about the water line) of the resultant thrust on the whole figure, acting at the centre of pressure of the figure, is equal to the sum of the moments (about the water line) of the resultant thrusts on the parts, acting at the centres of pressure of the parts. As a convenient part of the figure we take the strip between two horizontal lines at depths y and y + Ay feet below the water line. We know that its area can be expressed as X'Ay square feet, and the resultant thrust against it as zwy^'X'Ay lbs., if its centroid is at a depth yj feet. Let the depths of the centre of pressure of the whole figure and of the strip be y^ and y' feet. We have the equation Since the centre of pressure of the strip must, from the nature of the case, be a point on the strip, the limit to which yp' tends as Ay tends to zero is y. Hence the equation becomes Ja 174 CALCULUS [CH. XI Examples 1. In the case of a triangle with one side in the water surface, let the length of this side, BC in Fig. 75, be a feet, and let the distance of the opposite vertex A from BC be p feet. Since the triangles APQ, ABC are similar, we have a p Also Hence and therefore yG = -5P and S = g< 1 2 = 1 y^^{p-y)dy Jo P a/1 . 1 A 1 , yp=lp, or the depth of the centre of pressure is ^ the depth of the lowest point. 2. If one side of a rectangle is in the water surface the depth of the 2 centre of pressure of the rectangle is ^ that of the opposite side. o 3. If the centre of a circle is in the water surface, the depth of the centre of pressure of the immersed semicircle is -^ ir of the radius, or 0o89 lb of the radius approximately. [For the integration of. § 135 (ii).] Moments of Inertia 166. The motion of a very small body may be specified by the motion of one point of it. If wlbs. is the weight of the small body, and v feet per second its velocity, its kinetic energy is jr — ■u^ foot-pounds, where g stands for the number 32-2. If the 2 ff body is moving round a circle, so that the line drawn from the 165, 166] APPLICATIONS TO MECHANICS 175 centre of the circle to the position of the body turns with an angular velocity w radians per second, and the radius of the 1 tv circle is r feet, v = rut, and the kinetic energy is ^ — rV foot-pounds. We may think of a moving body in general as made up of small parts, the motion of each of which can be specified by the motion of one point in it. The kinetic energy of the body is the sum of the kinetic energies of the small parts. If the body is rotating about an axis we may take the points, by the motions of which the motions of the parts are specified, to describe circles about this axis, and the angular velocity will be the same for all of them. Let this angular velocity be ft) radians per second. Let the weights of the parts be ■Wi,jA)2, ...lbs. and the radii of the circles t-j, r^, ... feet. The kinetic energy of the body is „ ft)^ ( — rj^ H — -r^^ + ...) foot- pounds. The sum —r-^^+ —r^+ ..., or, as it may be written, 2 — r', is the measure in lb. -ft. units of a certain quantity called the ' ' moment of inertia " of the body about the axis. We shall suppose the body to be homogeneous. In finding the moment of inertia we may first suppose that the body is divided into thin sheets by means of a series of cylinders, having the axis about which the body turns as a common axis. All the parts between two cylinders, whose radii are r and r + Ar feet, will have velocities intermediate between rw and (r + Ar) co feet per second. If r' is some suitable number between r and r + Ar, and w' lbs. is the weight of all these parts, their w w' contribution to 2 — r° is — r''- 9 9 If a cubic foot of the body weighs o- lbs. this is the same as -r'^Ai!, where Au cubic feet is the volume contained between 9 176 CALCULUS [CH. XI the two cylinders. Then the expression for the moment of inertia becomes - Sr'^ A^j. 9 167. Let the body be a circular disc, of uniform thickness h feet and radius u feet, rotating about an axis passing through its centre at right angles to the planes of its faces. Then At) = A2ir ( r+^Ar ) Ar. (Cf.§58(fe).) Now let all the numbers Ar tend to zero, then Si-'^Au tends to a limit, which is /, 2Trhr . r^ . dr, or = Trfta*. The moment of inertia of the body is, in Ib.-ft. units, jr ir - ha*. 2 2 g Since inra^h lbs. is the weight of the body, say W lbs., we find that the IW moment of inertia is in Ib.-ft. units r; — a^. 2 9 168. The moment of inertia of a body about an axis W can generally be expressed in the form — k^ Ib.-ft. units, where Wlbs. is the weight of the body, g is the number 32 '2, and A is a number depending on the shape and size of the body. The length k feet is then called the " radius of gyration " of the body about the axis. The kinetic energy of the body is the same as if all the matter in it were condensed uniformly upon the cir- cumference of a circle, of radius k feet, with its centre on the axis about which the body rotates, and its plane at right angles to this axis. If V cubic feet is the volume of the body, Av cubic feet the volume of the part of it that is distant less than r + Ar feet from the axis, and more than r feet from the axis, then VA' is equal to the limit of the sum ^r'Av, the limit being arrived at by making all the numbers Ar tend to zero. The result obtained in § 167 is that the radius of gyration of a circular disc (radius a feet) about an axis passing through its centre at right angles to the planes of its faces is 5 a,J2 feet. 166-170] APPLICATIONS TO MECHANICS 177 169. In like manner we may define the radius of gyration (k units of length) of a plane figure about an axis in its plane. With the notation of § 163, the radius of gyration about the axis of y is given by the equation K'S o(?'idx. no. Aa an example, let the figure be a rectangle, let the lengths of its sides be 2a and 26 units of length, and let the axis of j/ be a straight line drawn through the centroid of the rectangle parallel to the sides specified by 26 (Fig. 76). Then we have S = 4a6, Y=26, a;o= -a, Xi = a, and ih^ah -/: 2bxHx=-^ J>a^, or J;2=-a2, Fig. 76. As a second example let the figure be a circle, of radius a units of length, and let the axis of y pass through the centre. Then we have and ira^k^ S = ira'^, y=2^{a^-x^, Xo=-a, Xi=a, 1 '4' = j 2x^J{a^-x^)dx=lwa^ (§ 135 (ii)), or k2=^a^. The radius of gyration of a plane figure about an axis in its plane is important in connexion with the theory of resistance of a beam to bending. We have found also in § 165 that the depth of the centre of pressure of a figure is determined by finding the depth of the centroid and the radius of gyration about the water-line. 12 178 CALCULUS [CH. XI TABLE OP STANDARD FORMS OF INTEGRALS. \x''dx = =-^, (A) J n+\' ^ ' j-dx = log,x, (B) L'^dx = - e"^, (C) I sin xdx = — cos x, (D) I cos xdx= sin sc, (E) I sec^ xdx = tan a;, (F) f .^ I _^. dx = ^in-^-, (H) i-^dx = -J^^~"l- W J a" + or a a 64 89 91 113 113 113 129 137 137 APPENDIX I. The Graph op a Rational Integral Function OF THE First Degree It is to be shown that all the points whose coordinates satisfy the equation y = mx + h he on a straight line. Let A, B, C be three of the points, named in such an order that x >x > x . ^ ' C B A Since y = mx + b, we have 2/c-2/b ys-^A Vc-Vf. = = = m, «'0-'^B '^B-^A "^C-^A and therefore, H y.>y^, y^>y^, and, iiy. ^'^'^ then we showed that the difference was less than £ if ^ was any number between and -r- . In general, by making h small enough is meant bringing h to lie between zero and some positive number h, if h is positive, or to lie between zero and — A, if A is negative. In the example k was -r— . We can put the general definition in the following form : — Let A be a variable which tends to zero, F {h) a function of h, U the limit to which it tends as h tends to zero. Any positive nupaber t, as small as may be wished, is chosen. Another positive number h is then found so that, if h is any number whatever between and k, or between and —k, the absolute value of F (A.) — L is less than c. When this can be done the right number L to be the limit is distinguished from all other numbers. We may show how the definition includes the results that ^ = when y = a, and ^^=1 when y = x, which were noted in § 17. First let y=a, a number independent of x. When x is changed to x + h, y is not changed. Writing f{x) for y we have f(x) = a, f(x + h) = a, and therefore f{x + n)-f(x) ^^ n for all values of h other than 0. In this case L=0. In fact we may take any positive number we please for k, and it is true that Q f(x + h)-f(x) ^^^ h f (x+ h)— f (x) if h has any value between and k or between and - k, for - — 1 ^ — is precisely zero. 182 CALCULUS Next let y = x. When x is changed to x + h, y ia changed ix) y + h, and, writing/ (a;) for y, we have f{x + h)-f{x) , h for all yalaes of h other than 0. In this case L=l. We ma; take any positive number we please for ft, and it is trne that 1 f(x + h)-f(x) h if h has any value between and k or between and - k. We may define in a similar way the limit to which a function F (a;) tends as x tends to a particular value a. The difference x-a may be positive or negative, but if, as a; - a tends to zero, F {x) tends to a limit L, the function F (a;) tends to the limit L as X tends to a. The definition may be written more at length as follows : — Let e be any positive number, as small as may be wished. If a positive number k can be found so that, for all values of x which lie between a and a + k, or between a and a-k, the absolute value of the difference F (a) - L is less than £, F (x) tends to L as a limit when x tends to a. The statement in § 13, that all the functions which we consider possess graphs, implies a certain limitation to which all the functions are subject. If y =f(x), and f{x) has a graph, then when x is changed to a; + Aa; and y to y + Ay, it is imphed that Ay tends to zero as Aa; tends to zero. According to the above definition this is the same thing as saying that, as x tends to any value a,f(x) tends to a limit, which is/(a). A function which is subject to this limitation is said to be "continuous.'' The only kind of discontinuity which is met with in elementary work is the kind which occurs if there is a value of x for which the function/(a;) cannot be calculated, because the formation of the expression for f{x) would require division by zero. For example, the function - is not continuous when x = 0. The possibility of such values implies a restriction upon the generality of some theorems. The theorem of § 56 can be proved to hold for any function f{x) if the derived function /' (a;) is continuous II] APPENDIX 183 for all values of x between a and b inclusive. The theorem rb (§ 154) that the definite integral / F'{x)dx, considered as the Ja b limit of a sum such as % F' (a;) Ate, is equal to F (6) - F (a), can be a proved to hold for any function F (x), if the derived function F' (x) is continuous for all values of x between a and b. In elementary work these restrictions are always understood. We prove here a series of four theorems concerning limits. AU of them are nearly obvious, but it is satisfactory to prove them from the definition. (1) If, as h tends to zero, F (A) tends to L as a limit, then aF (A) tends to osL as a limit, a being any constant. Let ri be any positive number, as small as we please. We know that a positive number k can be found so that, when h lies between and Jc, or between and — k, the absolute value of F (A) - L is less than i;. Let e = ai; or —arj according as a is positive or negative. Then e can be any positive number, as small as may be wished. The absolute value of osF (A) — aL is less than t. Therefore aL is the limit of aF (A). A special case of this theorem was used in proving the first Bule of difierentiation (§ 21). (2) If, as A tends to zero, Fj (A) and Fj (A) tend to Lj and Lj as limits, then Fj (A) + Fj (A) tends to Lj + Lj as a limit. Let 17 be as before. We know that a positive number ki can be found so that, when A lies between and k^, or between and — ki, the absolute value of Fi(A) — Lj is less than i;. We know also that a positive number k^ can be found so that, when A lies between and k^, or between and —k^, the absolute value of Fj (A) - La is less than rj. Let A be a positive number less than either k^ or k^, and let 2r] = e. Then e can be any positive number, as small as may be wished, and we know that the absolute value of Fi (A) -L,+ F, (A) - L, or of {Fi (A) + Fj (A)} - {L, + L,) 184 CALCULUS is less than i, when h lies between and k or between and - k. Hence Fj (h) + Fj (h) tends to a limit, and Lj + Lj is that limit. A special case of this theorem was used in proving the second Rule of differentiation (§ 21). When there are three terms Fj {h), Fg (h), Fj (h) and their limits are Lj, Lj, Lj, we have limit of {Fi (A) + Fj (h)} = (Lj + Lj). Now Fi (h) + Fa (A) + F3 (h) = {Fi (A) + Fa (A)} + F3 (h). The sum of three terms is expressed as the sum of two which have limits (Lj + Lj) and L3. Hence limit of Fi (h) + Fj (h) + Fg (A) = Lj + L^ + Lj. In the same way the theorem may be proved for a sum of n terms if n is any whole number. (3) If, as h tends to zero, Fi (h) and Fj (h) tend to L, and L-j as limits, then Fi (A) . Fj (h) tends to LjLj as a limit. Let Fi (h) = Li + X, Fs (A) = La + Y. We know that X and Y tend to zero as a limit. Now F,(A).Fa(A) = (Li + X)(La + Y) = LiLj + LiY + LjX + XY. When k is chosen as in Theorem (2), the absolute value of XY is less than rf, and this is less than 1; if t; is less than 1. Hence the product XY tends to zero as a limit. By Theorem (1) LjY and LjX tend to zero as a limit. Therefore by Theorem (2) LjY + LaX + XY tends to zero as a limit. Hence F, (A) . Fj (A) tends to LjLj as a limit. A special case of this theorem was used in proving the Rule for diflFerentiating a product. Just as in Theorem (2) the result may be extended to the product of n factors if n is any whole number. Il] APPENDIX 185 (4) If, as h tends to zero, F {h) tends to L as a limit, and if L is not zero, -,,, tends to - as a limit. F(A) L Let r) be any positive number, as small as may be wished. We know that a number k can be found so that, when h lies between and k, or between and — k, the absolute value of F (A) — L is less than -q. Now 1 1 _ F (A) - L L~F(A)~ LF(A) ■ To fix ideas let L be positive, and let F (A) = L + X. When h is as above, the absolute value of X is less than 1], and the absolute value of F (li) is greater than L — 17. Hence 11 -n the absolute value of 777 is less than ; — r . If we put L F{A) '-(L-i;) e for this, £ can be any positive number, as small as may be wished. We have therefore proved that, when e is chosen, a positive number k can be found so that, when h lies between and k, or between and — k, the absolute value of ... F (II) L is less than e. A special case of this theorem was used in § 25 in proving the Kule ^ T- = 1. ax ay In the investigation of limits it is often convenient to in- troduce a set of numbers Mj, ttj, ... m„, ... with some rule by which the »ith number of the set, m„, can be expressed in terms of the integer n. Such a set is called a "sequence." The sequence may have a limit L. Let any positive number e, however small, be chosen, and let a number N be found so that, when n is any integer greater than N, the absolute value of m„— L is less than £. Then L is the limit of the sequence. For example, if 1 + - 1 , L is e, as will be proved below. Let e be as above ; and suppose that we can find an integer N so that, if m and 186 CALCULUS n are any integers greater than N, the absolute value of m„ — u„ is less than e, then we know that the sequence has a limit. Many theorems concerning limits are proved by using two sequences aii, Kj, ...a!„, ... and y^, ys,---yn. If '^^ can show (i) that as n increases x^ increases and y^ diminishes, (ii) that every one of the numbers y is greater than any one whatever of the numbers x, (iii) that the difference y» — aj„ can be made as small as we please by increasing n sufficiently, we can infer that both sequences have the same limit. When it is said that, as n increases, £b„ increases, it is not meant that necessarily X2>Xi, x^>x^, and so on. What is meant is that, whenever a change occurs as n increases, the change is an increase. Similarly for the y'&. Now let e be any positive number, as small as may be wished, let k be an integer such that y — x < e, and let m and n be any integers greater than k. For clearness take m to be greater than n. Then ym^Vn^yK ^'i'^ ^m,'^ ^n"^ ^^t ^iid therefore £c„— a5„ < e and yn~ym< «■ This proves the statement. We may illustrate this method geometrically by taking the it's and the t/'B to be the numerical measures of distances of points in a straight line from a fixed point in the line. To fix ideas, we are supposing all the numbers x and y to be positive. As the suffix increases the X point always Xi Xs Xj X4 Y4 Y3 Y2 Yi • 1 1 1- \: i 1 -I- Fig. 78. moves to the right, and the Y point to the left (Pig. 78). When n is large enough the distance between an X point and the corresponding Y point is very short. All the subsequent points of both sets are crowded into this very short length, and yet every Y point is to the right of all the X points. As an example of the application of this method let u^, u^,...Un,... be a sequence of positive terms, let w„ increase as n increases, and let it be known that no term is so great as a certain number A. The sequence has a limit which is either the number A or a number less than A. We form two new sequences so^, x^,... and yuVii---- Let ajj be itj and yi be A. Il] APPENDIX 187 Consider the number half way between x^ and y^, it is Mi + n(A — Ml), denote it by Bj. Either some number of the sequence u is greater than Bj or else no number of the sequence u is greater than Bj. In the first case let scj be Bj and let 2/2 be the same as y^ ; in the second let x^ be the same as x^ and let 2/2 be Bj. Choose in the same way a number B^ half way between x^ and y^ and form with it the numbers Xg, y^. At every step we halve the difierence 2/1 — a^i, and after n—\ steps we have yn — ^n= t)n-i^ ' which can be made as small as we please by increasing n sufficiently. Now after n—l steps there are numbers of the sequence u which are greater than x^ but no numbers of the sequence u which are greater than y„. The numbers of the sequence u which lie between x^ and y^ are those whose sufiixes exceed some particular integer N. It appears therefore that by taking N great enough and I and m to be integers greater than N we can make the absolute value of the difference M; — u^ as small as we please. Hence the sequence u has a limit. We may illustrate this process geometrically. Finding Bj is bisecting the length XiYi. Finding B2 is bisecting the length XjBi or BiYi accord- ing as there are not or are to the right of Bj points specified by some u. To O Xi X, Yu ' ' Bi K Yt Fig. 79. fix ideas suppose that BjYi is bisected. Then the next step is to bisect BiBj or B2Y1 according as there are not or are to the right of B2 points specified by some u. We very soon find all the values of «„ for high values of n con- fined between two points which are very close together, and we can bring them as close together as we please, thus crowding all the values of «„ for high values of n close to one point — the limit to the left of which all the points corresponding to values of «„ lie. In like manner if M„ always diminishes as n increases, but remains always greater (algebraically) than some fixed number A, the sequence has a limit, which is either A or a number algebraically greater than A. 188 CALCULUS III. Indices and Logarithms The definition of lognja; by the equations 10" = 03, y = log^^x is in some ways incomplete. The question might be asked ; How is it known that to a given number x there corresponds a number y which makes 10* equal to a; ? If a; happens to be 1 or 10, or ^ , or 100, or ^^j or any positive or negative integral power of 10, 2/ is known. If x happens to be the square or cube root of 10, or any other root say the g'th root, or if it is the pih. power of the g'th root, y is known. But some numbers are not powers of roots of 10. If for instance x were 2, no fraction - could be y. If 108 were 2, 10" would be 29, and 5^ would be 2«-*, an odd number would be equal to an even number, which cannot happen. Then it might be said that, for that matter, the square in root of 10 and many other numbers are not fractions like -, and that when we speak of 10" as being equal to 2 we must mean y to be a number of this sort. But, if this is said, the question may be asked : What does 10" mean if y is not an integer or a fraction! 10" was defined by the law of indices for integers and fractions, but it has not been defined for any other numbers. The question can be answered by taking proper account of the true meaning of such numbers as 10* "irrational" numbers, as they are called, to distinguish them from the ordinary " rational " numbers. Integers, and fractions of which the numerators and denominators are integers, together constitute the class of numbers called rational numbers. Now when we find by the ordinary arithmetical process th-e square root of 10 correctly to a number of places of decimals we really show successively that the numbers 4, 3"2, 3'17, 3'163, 3'1623 and so on have their squares greater than 10, but that the numbers 3, 3-1, 3'16, 3'162, Ill] APPENDIX 189 3'1622 and so on have their squares less than 10. That is to say, we gradually sort the rational numbers into two sets : a set of which every one has its square greater than 10, and a set of which eveiy one has its square less than 10 ; and the square root of 10 means a number which is less than every number of the first set and greater than every number of the second set. An irrational number like 10* may properly be said to be " known " whenever we have a process for determining, in regard to any rational number, whether that rational number is greater or less than the irrational number in question. If 2/ is a rational number of the form — ^ — , where m and n are integers, 10^ is a known irrational number. If y (supposed positive) is any known irrational number, or any rational number that is neither an integer nor of the form — qj — , and if a and b are any two rational numbers of this form, such that a>2/ but 610'', 10" -y. If however 10''>y> IC", we choose a smaller number a from the a set and a larger number 6 from the b set. It may happen that every number of the set 10" is greater than y and every number of the set lO"" is less than y. In this case 10" is equal to y. But, if this is not the case, either (i) y is greater than some number of the set 10", or (ii) y is less than some number of the set 10^ We can therefore sort the rational numbers such as y into two sets : those which are greater than 10" and those which are less. There cannot be two rational numbers y^ and y^ which are in neither set, because we can take a and b so near together that 10"— 10* is less than the absolute value of yi — y^. If there is one rational number which is in neither set 10" is rational and is equal to that number. If 190 CALCULUS every rational number is either in one set or in the other we have a process for determining, in regard to every rational number, whether it is greater or less than 10". That is to say 10* is a known irrational number. We have shown how to define 10" for any positive value of y, and it is easy to prove from the definition that 10" obeys the laws of indices for all positive values of y. We define 10 ~", for y positive, as ^ . The theory here described may be compared with the process suggested in Oh. VI for forming a Table of logarithms by finding, correctly to as many places of decimals as may be wished, the square root, fourth root, and so on, of 10 and of integral powers of 10. We have seen how, beginning with a positive known index y, rational or irrational, we can define 10", and we have also seen how to extend the definition to negative values of y. We have now to show that, if we begin with a positive number x we can define a number y which is such that 10" = a;. We may suppose x to be greater than 1; for, if x were less than 1, - would be greater than 1, and if we could define a number y' so that 10"' is equal to - then y would be — «/. Take any rational number C, we know X how to define 10°, and if it is not equal to x it must be either greater or less. We can sort out all the rational numbers into two sets, those which make lO'' greater than x and those which make 10° less than x. There may be one rational number C which is in neither set. There cannot be more than one. If there is one, that one is the required y, and y is rational. If there is not one, then there is an irrational number y which is such that (i) for every rational number a, which is greater than y, 10">a!, and (ii) for every rational number 6, which is less than y, 10* < X. Then, by the definition of 10", we have 10" = a;. Ill, IV] APPENDIX 191 IV. The Exponential Limit / - It is to be proved that, as h tends to zero, (1 + -) tends to a limit. We begin by taking h to pass through such a sequence of values that r is always a positive integer n, and we show that, as n increases, (l H — 1 increases*, but is always less / 1 \"+' / 1\" n+l if fl + -^)">l+i, and this is the case if \ n+\J n or again if n j ( 1 + =- j — 1 [ > 1. 1 We put „ = (^1 + _!_)», so that «" = (!+ =■ ) , V TO + 1/ and therefore (as" - 1) .(?i + 1) = 1. Then we have to prove that „(„»+! _l)>(w + l)(a«._l). Now this is true for any value of a greater than 1. In fact, we have a-'H-i-l = (a-l)((i« + a''-i + a"-2+ ... +a^^a^\), and a"- l = (a-l)(a"-i + a""''+... + (i' + a+l), * Of. G. Chrystal, Algebra, Part II. p. 77 (Edinburgh, 1889). 192 CALCULUS so that the inequality we have to prove is n(a-\)d^ + n {a- 1) (a«-i + a"-= + ... + a+ 1) > (re+ 1) (a- 1) («»-' + «»-==+... +a+ 1), or n(a-l)a''> (a-l)(a''-' + a''-='+... +a+l). On the right-hand side we have n terms, each of which is less than (a— l)a'', a being greater than 1, and therefore the inequality is proved. Again, we show that, as n increases, I 1 1 diminishes. (>-r>('-^.)""". if I \ n+x > {'-IJ ('-.4-0 n+l orif l_i<(l_ 1)», n \ n + lj and this is the case if \ n + l/ n n+l or again if w |l - ^1 - ^^-j-^j " j "* ^• 1 We put 6 = (l__l^)«. so that 6" = 1 = , and therefore (1 - 6") (?i + 1) = 1. Then n + l we have to prove that w(l -6"+i) < («+ 1)(1 - 6"). IV] APPENDIX 193 Now this is true for any value of b less than 1. In fact we have l_6«+i = (1 -5) (1 + 6 + b^+... +6»-2+ 6"-^+ 6»), and 1-6" =(1 - 6) (1 + 5 + 6^ + ... +6«-2 + 6"-'), so that the inequality we have to prove is m(l-6)(l +6 + 6''+...+6"-^+6"-i) + n(l-6)6'' ^ (n + 1) (l - b) (I +b + b" + ... + 6"-="+ S"-!) or 7i(l -5) 5" < (1 -6) (1 + 6 + 6^ + ... + 6"-2 + J"-!). On the right-hand side we have n terms, each of which is greater than (1—6)6", 6 being less than 1, and therefore the inequality is proved. Kow,U„ = 2, (l-l)-.(l-l)"".4. ('4)"^('-s)"-('4.)'. which is less than 1 when n is greater than 1, and therefore ('4)-<('-r- It follows that (1 + -) <4. We have therefore proved that, as the integer n increases, ( 1 + - ) increases, but remains always less than 4. It therefore tends to a lijnit (see pp. 186, 187). We call this limit e. Next we take the case where h tends to zero in such a way that - is always positive, but not necessarily an integer. If x is positive, h also is positive, and, as h is diminished towards zero, h passes through all positive numbers, rational or irrational, which are less than some chosen number. Similarly if x is negative. We write z for j-. Then, whatever positive non- L. c. 13 194 CALCULUS integral value we give to z, there is an integer n such that « lies between n and n + 1, and, as h tends to zero, n increases / 1\^ / 1\" / 1\"+' indefinitely. Now (l+-j >(l + -j but <(!+-) . Also 11 1 / W !+->!+ ^ but < 1 + - . Hence ( 1 + - 1 lies between 2 n+ 1 n \ zj ( 1 + i ) and ( 1 + - ) , i.e. between (1 + ) and n + - j . As the integer n increases indefinitely, both these expressions tend to the same limit e, and therefore tends to e as a limit. (-^r Finally, let - be negative, and write a for - . Put % = — %', then »' is positive, and as h tends to zero, »' increases in- definitely. Now As »' increases_ indefinitely, this expression tends to « as a limit. It is not relevant to the argument, although it is true, that the limit e in question is the same as that to which the expression ,,11 1 1 + ^ + 21 + r! + '"+iU tends as the integer V, increases indefinitely. Here ft I means the product 2 X 3 X 4 X ...(ft - 1) X %. This result may be used to compute e approximately, but it is not necessary to know it in order to prove that I 1 + - J tends to a limit. A parallel case is presented by the number ir. We may prove that the area of a regular polygon of 2" sides inscribed in a circle tends to a certain limiting area as the integer n increases indefinitely, by showing that as n, increases the area increases, and yet that it remains always less than the area of the circumscribed square ; and we may call the numerical measure of the limiting area in^ when the numerical measure of the radius is r. It is IV] APPENDIX 195 not relevant to the argument, although it ia true, that the number v Is also the limit to which the expression tends as h increases indefinitely, but this result might be used to compute ir approximately. For the calculation of e we use the theorem of § 54, viz. : that, if fix) vanishes when x=a and when x = h, f (x) vanishes for some value of x between a and h. We use also the result -rr- = e". ax We know that e > 1. Write Rj for e — 1. Then put /i(a:)=e-e^-(l-a;)Ri. We see that f (x) vanishes when a3= and also when x=\, and therefore // (as) vanishes for some value of x between and 1. But f-l (£c) = — e^+ Ri. Hence Rj is equal to the value of 6* for some value of x between and 1. We see that Rj > 1, and therefore that e > 2. Put R2=Ri-l=e-2, and /s (a:) = e-e="-(-l-a;)e^-(l-a;/R2. Then/s (x) = when x = l and also when aj = 0, and therefore fi {^) vanishes for some value of x between and 1. But f^ {x) = - e" + e' - {\ -x) e" + 2 {l~x) R2= 2 (1 -a;) (Rj- ^e"). Hence Ra is equal to the value of jj ff" for some value of X between and 1. This value of x is, of course, not the value which made Ri = e'". We see that R2>h> and, since e = 2 + R2, that e>2-5, Also we see that R^ < ^ e, and therefore that e < 2 + - e, or e < 4. Put R3=R2-i = e-2-l, and j,{x) = e~e'-{l -x) e"- -^{l-xYe~{l -xf R3. 13—2 196 CALCULUS Thenyg (x) = when a: = 1 and also when x = 0, and therefore /a' (x) vanishes for some value of as between and 1. But fa {x) = -e'+ ^-{l~x)e' + {l -x)e'-l (1 -x)' ef" + 3 {I -xf R, =Hi-^y{^s-^,^). Hence R3 is equal to the value of ^ e" for some values of x between and 1. We see that R3 > ^ but < ^ «• Hence e>2 + -^ + -z^, or=, but 115 5 8 e<2+^+ ^e, or^e<-, ore<3. Thus e lies between -^ and 3. Put R^=R3_| = e_2-i-l, and f,(x) = e-e"-(l -00) e" -I (1 - a;)V- ^ (1 -»)» e«- (1 -a;)* R,. Then f^ {x) vanishes when a; = and also when x=l, and therefore ^' (x) vanishes for some value of x between and 1 . But /:{x) = -~(l-xye' + i{l-xyR„ for all the remaining terms cancel, or we have /;(.)=4(i-.)3(r,-^^^.^). Hence R- = ^rr e" for some value of x between and 1. We 24 R,>lbut2 + ^ + g+2j, „ 1 1 1 23 8 ^^2 + 2+6 + 24' "•■ 24'^3- ,. , 65 J 64 Thus e lies between ^ and no • IV] APPENDIX 197 We can proceed in this way, getting closer and closer approximations to e. Put in general g 1 1 1 1 * * 2 2x3 2x3x4 "• 2x3x4x ...(A-1)' and /, (^) = e - e- - (1 - a;) e- - .^i^V - ll^V -- 2x^3:.ir-i) -^-(^-)^"-- Then f^ (x) vanishes when x = and also when a; = 1, and therefore /j.' (x) vanishes for some value of x between and 1. But, just as before, we find -k{l -Xf-- |r, - 2^3x...(;i;_l)j ' and therefore Ri, = -= — = r for some value of x between * 2 X 3 X ...A and 1, or 1 _J_ 1 1 e' *~ 2"^ 2x3 '^2x3x4"^'""^2x3x...(A;-l)"^2x3x...A' for some value of x between and 1. Thus e lies between - 1 1 1 1 2 2.3 2x3x ...(/fc-1) 2x3x...A' and 2 + TT + t-— ^ + . . . + 2 2.3 2.3...(A-1) 2x3x...A' for e* < 3 when a; < 1. "We see that we can calculate e to any desired degree of accuracy. 198 CALCULUS V. The Mensuration op the Circle and the Radian Measure op Angles. Let the radius of a circle be r units of length. Let squares be circumscribed about and inscribed in the circle. Pig. 80 shows at once that their areas are ir" and 2»^ units of area. The straight ILues drawn through the centre to bisect the sides of the inscribed square, meet the circumference in 4 points, which, together with the 4 comers of the square, make up the 8 corners of an inscribed regular octagon. Fig. 80 shows that the area of Fig. 80. the octagon is greater than that of the inscribed square. In like manner, by bisecting the sides of the octagon, we may find the 16 comers of a regular inscribed polygon of 16 sides, and the area of this polygon is greater than that of the octagon. We may proceed to inscribe regular polygons of any number of sides, the number being a power of 2, and we see that, as the number of sides increases, the area of the polygon always increases. But, as no point in any inscribed polygon can be outside the circle, the areas of all the inscribed polygons are less than the area of V] APPENDIX 199 the circumscribed square. Hence the number of units of area in the area of the regular polygon of 2" sides inscribed in the circle increases as n increases, but never becomes so great as it"- It therefore tends to a limit (see pp. 186, 187). This limit is defined to be the number of units of area in the area of the circle. Let regular polygons of 2" sides be inscribed in two circles whose radii are j-j, r^ units of length. The two polygons are similar figures and their areas are proportional to the areas of the squares described upon corresponding lines in the two figures, and therefore the measures of their areas are proportional to the numbers r^^ r^, and the areas are expressible as j^^^ and f^r} units of area, where f^ is a number which depends upon n, but not upon rj or i\. As n increases, the number /„ tends to a limit. This limit is called ir. The area of the circle, of radius r units of length, is lefi units of area. Let the length of the perimeter of the inscribed regular polygon of 2" sides be p units of length, and let the length of the perpendicular drawn from the centre to any side of this polygon be q units of length. The area of the polygon is ^ pq units of area. Now, as n increases, q tends to a limit, which is r. Hence p tends to a limit which is 2irr. The limit to which the length of the perimeter of the polygon tends is defined to be the length of the circumference of the circle. We learn that it is litr units of length. When we inscribe in the circle a regular polygon of 2" sides, we divide the part of the plane which lies within the circle into 2" congruent sectors, and the circumference into 2" congruent arcs, and the angles subtended by these arcs at the centre are aU equal. The area of each sector is -^ units of area, the length of each arc is -^ units of length, and the measure of each angle is 4 2i nglit angles. 200 CALCULUS Now let AB be any arc of the circle, and let A be one vertex of an inscribed regular polygon of 2" sides. If, for any value of n, B is another vertex, there may or may not be other vertices on the arc AB. We can include both these cases in the same statement by taking the number of intermediate vertices to be m— 1, where m may be 1 or may be greater than 1. Then the length of the arc AB is -^ 2irr units of length, the magnitude of the angle which it subtends at the centre is 4 ^ right angles, and the area of the corresponding sector is -^ irr^ units of area. If, however, B is not a vertex for any value of n, we may suppose that, when the polygon has 2" sides, there are tn vertices on the arc AB between A and B (A not counted as one). Then the magnitude of the angle subtended by the arc AB at the centre lies between 4 ^ and 4 right angles, and the end B of the arc AB lies between two points P and Q which are at the ends of arcs AP and AQ of lengths ^ lirr and — ^j — ittr units of length, also the areas of the sectors standing on the arcs AP and AQ are m J ™ + 1 » • . c ■^ Trr and „„ trr units of area. 2" 2" We write a;„ and y^ for the numbers -^ and „„ , and con- sider how the numbers a5„ and y„ are altered when n is increased. When n is changed into w + 1, the number of sides of the inscribed polygon is doubled, and new vertices are introduced. These new vertices lie on the bisectors of the angles which are subtended at the centre by the sides of the polygon of 2" sides. We name these angles the first, the second, and so on, beginning with that angle of which one side passes through A. If the bisector of the (ni+ l)th angle does not cut the circumference at a point on the arc AB, m is changed to 2m; if it does, m is V] APPENDIX 201 changed to 2m + 1. When n is changed to w + 1 and m to 2m, a!»+i = a'». but 2m + 1 m + 1 When n is changed to n + 1 and m to 2m + 1, _ 2m + 1 m , , 2m +2 m + 1 but y»+i= 2^+1 = ~2^' "'' Vn-n =!/«.■ If we keep on doubling the number of sides of the polygon, at every step one of the two numbers x^, y^ is changed and the other is not. When x^ is changed it is increased, when y^ is changed it is diminished. But, since the magnitude of the angle subtended by A B at the -centre always lies between 4a;„ and 4y„ right angles, all the numbers x^ are less than any one of the numbers 2/„. Further yn—^n — T;^ *nd this can be made as small as we please by increasing n sufficiently. It follows that, as n increases, a3,i"and y„ tend to a common limit (see p. 186). Let us denote this limit by 75— . Then the angle which AB 2a subtends at the centre is — right angles, the lengths of the arcs AP and AQ tend to one and the same limiting length, which is ar units of length, and the areas of the sectors standing on AP and AQ tend to one and the same limiting area, which is ^ ar' units of ii area. We define this limiting length and limiting area to be the length of the arc AB and the area of the sector standing on this arc. It appears that, whether B is a vertex or not, there is a number u. which is such that (i) the angle which the arc AB subtends at the centre is — right angles, (ii) the length of the 202 CALCULUS arc AB is ar units of length, (iii) the area of the sector standing on this arc is -^ ar' units of area. The number u measures the angle in terms of a certain angle chosen as unit. This unit angle 2 is the " radian." Its magnitude is — right angles. VI. Tkigonometeic Limits. If the lengths of an arc and chord of a circle are I and x units of length, then, as the ends of the chord approach to coincidence, - tends to 1 as a limit. Let the chord PQ be bisected in N, let the straight line joining the centre C to N meet the circle in A, as shown in Fig. 81, and let the tangent PT meet the straight line CA in T. Let the angle ACP be a radians, and the radius of the circle r units of length. Then the lengths of the chord PQ and arc PQ are 2?" sin a and 2ra units of length. We have therefore to prove that, as a tends to zero, -: tends to 1 as a limit. sm a The lengths of PN, the arc PA, and PT are r sin a, ra, and r tan a units of length, and the areas of the triangle PCA, the sector PCA, and the triangle PCT are 5 r' sin a, 5 r\ and ^j^tana units of area*. Hence Fig. 81. or we have sin a < a < tan a. 1 < —. — < sec a. sina * Cf. E. W. Hobson, Trigonometry, Oh. VIII Vl] APPENDIX 203 When a tends to zero sec a tends to 1 as a limit, and therefore -: — , which lies between 1 and sec a, tends to 1 as a limit, sin a We may deduce the result that when o tends to zero tends to a zero as a limit. We have sm2 a= (1 - cos a) (1 + cos a), and therefore 1 - cos a sin a 1 a a 1 + cosa Now when a tends to zero, cos a tends to 1 as a limit, and 1 + cos a tends to 2 as a limit, and therefore the limits of the three factors on the right-hand side are 1, - , 0. This proves the result stated. By means of these two results and the addition equation for the sine, sin (A + B) = sin A cos B + cos A sin B, we may differentiate sin x. We have sin (x + h)=Bmx cos ft + cos a; sin fe, and therefore sin {x+h)- sin a; _ sin ft eos x-(l- cos ft) sin x h ~ ft sin ft . 1 - cos ft = cos a; — 1 sm x ; . ft ft The limit to which the right-hand member tends as ft tends to zero is cos X. In like manner we have the addition equation for the cosine, cos (x -I- ft) = cos a; cos ft- sin X sin ft, and therefore cos (x-l-ft)-cosx _ sin x sin ft -f cos a; (1 - cos ft) ft ~ ft sin ft 1 - cos ft = - sm X —^ cos X ; , ft ft and, as ft tends to zero, the right-hand member tends to - sin x as a limit. This method of proof appears much shorter than that given in Ch. VII., but, to render it complete, we should require to prove the formulca for sin(A-t- B) and cos (A -|- B) for all values of A and B, positive and negative. This is, of course, done in books on Trigonometry. The method used in Ch. VII. shows that sin x and cos x can be differentiated without proving these formula. 204 CALCULUS VII. Mechanical Units. In the Chapters of this book, containing applications of the Calculus to Mechanics, the units employed are those of the so-called " British Engineers' System." In this system of units the fundamental quantities are force, length, time. Mass is a derived quantity. The unit of "force is the force required to support in London a body which weighs 1 lb. in a common balance. This is the "force of 1 lb.'' The units of length and time are 1 foot and 1 second. The unit of mass is adjusted so that the force of 1 lb. acting upon a body whose mass is 1 unit of mass may produce in it 1 unit of acceleration ; it is therefore the mass of a body which weighs 32-2 lbs. in a common balance. INDEX [The numbers refer to the pages.] Abscissa, 5 Acceleration, 34, 80 Addition equation, 203 Angular velocity, 100, 175 Approximations, 48-46, 92, 114, 139, 138 Arbitrary constant, 61 Arcs of curves, 109, 146, 164 Area of a plane figure, 73, 163 ; of a surface of revolution, 152, 164 ; under a curve, 58, 69, 157. Average velocity, 6, 7, 19 Axes, 2 Convexity and Concavity, 35 Coordinates, 8 Cosine, 101 Cubic equations, Approximate solu- tion of, 14 Curvature, 149 ; Eadius of, 150 Delta, 17 Derived function, 18 Differential coefficient, 18 ; second, 33 Differentiation, 13 ; Bules of, 23, 28 Distance-time graph, 7 Burrau, C, 103 Calculation, of ir, 159 ; of e, 160, 195 Centroid, 169 Chemistry, Applications to, 93 Chrystal, G., 191 Circle, Equation to a, 88 ; Area of a, 57, 199; Arc of a, 109, 199; of curvature, 151 ; Badius of gyration of a, 177 ; Mensuration of the, 198 Cone, Volume of a, 76 ; Area of sur- face of a, 152 Electricity, Applications to, 95, 121 Equation to a curve, 38 Equations, Number of real roots of, 54 Error. See Approximations Expansion, Coefficients of, 45 Exponential function, 89-91 Exponential limit, 88, 191 Falling body, 9, 80, 96 Function, 16 ; Differentiation of a function of a, 28 206 INDEX [The numbers refer to tlie pages.} Gradient, 8, 12, 18, 148 ; Sign of, 8, 47, 49, 61 Graph, 6 Graphs, Examples of, 14 ; of special functions, 5, 10, 85, 90, 104, 105, 107, 108, 136 Gravity, Acceleration due to, 80 ; Motion of a projectile under, 81 ; Centre of, 166 Gyration, Badius of, 176, 177 Harmonic motion. Simple, 116 ; Damped, 120 Hemisphere, Centre of gravity of a, 167 Hobson, E. W., 202 Homogeneity, 167 Hyperbola, 42 Indices, 27, 188 Integral, 62 ; Definite, 70, 157 ; Indefinite, 71 Integration, Methods of, 63, 64, 125- 143 ; Approximate, 158-160 Intermediate value, Theorem of, 52 Irrational numbers, 188 Limit, 11, 21, 180-187 Linear function, 26 Logarithms, 84-89, 188 Maxima and Minima, 46-51, Ex- amples of, 51, 114, 115 MeUor, J. W., 93 Mensuration, 55, 59, 73, 75, 152, 198 Moment, of a force, 166 ; of inertia, 175 Negative numbers, Bepresentation of, 3 Normal to a curve, 41 Ordinate, 5 Origin, 2 Oscillatory motions, 115-121 Parabola, 10, 38 ; through three points, 39, 42, 71, 159 ; Area under a, 68, 71 ; as path of projectile, 83 ; Length of arc of a, 149 ; Curvature of a, 152 Partial fractions, 127 Farts, Integration by, 132 Perpendicularity, Condition of, 40 Pressure, Centre of, 173 Product, Differentiation of a, 25 ; illustrated geometrically, 45 ; Limit of a, 184 Pyramid, Volume of a, 78 ; Centre of gravity of a, 168 Quadrants, 4 Quotient, Differentiation of a, 31 Badian, 99, 202 Badius, of curvature, 160; of gyra- tion, 176, 177 Bate of change, 19 Bational function, 31 Bational integral function, 26 Bectangle, Centre of pressure of a, 174 ; Badius of gyration of a, 177 Beduction, Formula of, 133 B^sistance to motion, 96, 118 Botation, 100, 175 Secant (cutting line), 12 ; (trigono- metrical), 106 Second differential coefficient, 33 Semicircle, Centroidof a, 170; Centre of pressure of a, 174 Sequence, 185 Sigma, 163 INDEX 207 [The numbers refer to the pages.'} Simpson's Rule, 159 Sine, 101 ; Inverse, 136 Speed, 19 Sphere, Volume of a, 76 ; Area of surface of a, 154 Spring, Motion of a body attached to a, 116-121 Squared paper, 2, 158 Standard forms of integrals, 64, 178 Straight line. Equation to a, 39, 180 Sugar, Inversion of cane, 93 Sum, Differentiation of a, 24 ; Inte- gration of a, 63 ; Definite integral as limit of a, 160-163 ; Limit of a, 184 Tables, of certain powers of ten, 85 ; illustrating the exponential limit, 87 ; of the exponential function 90 ; of the sine, 104 ; of the tan- gent, 108 Tangent (touching line), 12, 20, 41, 148 ; (trigonometrical), 106 ; In- verse, 136 Thomson, J. J., 95 Thrust, 171 Trapezium, Area of a, 56 Triangle, Area of a, 55, 74 ; Centroid of a, 170 ; Centre of pressure of a, 174 Trigonometric functions, 101-118 ; Inverse, 135-137 Trigonometric limits, 202 Uniform motion, 6 Units, 204 Variable, Change of, 29, 64^67 Velocity, 7, 8, 11, 19, 80 Velocity-time graph, 34 Volume of a solid figure, 75, 163 Weight, 166 Cambtitigt : PRINTED BY JOHN CLAY, M.A. AT THE UNITEB8ITT PRESS.