50 INFINITESIMAL CALCULUS Schneider, Sehlomilch, and others. The most recent and complete tables are those of Mr J. W. L. Glaishcr, already referred to ( 131). 161. The values of some definite integrals can be best determined by transforming them into infinite series. This statement will be illustrated by one or two examples. - n (1) To find / -^ dx. Jo 1-35 Here, when x is less than unity, log a / ?!L?L fa = Jo l-x CA (2) In like manner it can be shown that (3) Again, to find 1 -. . (1+a?) log Replacing i by its development, we get l+X Consequently (Ex. 4, 148) = log (2-a)(4-q) . . . = log (ia - %?)(& - a*) = log tan ^T by a known formula in trigonometry. 162. Conversely,, an infinite series can in many cases be trans formed into a definite integral, and thus evaluated. For example, suppose 1 r l Here, since = / x- n dx, we have 2n + 1 y S = f l dx(l+x*-x*-xG+ . . O- y<) In like manner we get Again, the series 1 _ o 1-a; 8 "" 4 2 1 + 1) . . . (p + m + n) + &C. can be represented by a definite integral.
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Here r(7i+i)y ( 153) ; 1.2.3... TZ./O 1- We now proceed to give a few applications of the calculus to geometrical problems. A reas of Plane Curves. 163. If a plane curve be referred to rectangular axes of coordinates, the area between the curve, the axis of X, and two ordinates corre sponding to the abscissa a and b is represented by the definite integral / ydx. Hence if y = <j>(x) be the equation of the curve, the area in question is denoted by /b <(>(x)dx . From this result it follows that every definite integral may be represented by an area. And it is seen at once that all the examples hitherto considered admit of geometrical interpretation. In the above formula the ordinate is supposed positive for all points of the curve between the limiting abscissa . The modifica tion when the curve cuts the axis of x can be j-eadily supplied. Ex. 1. Let the curve be an ellipse, represented by the equation o-2 i/2 B ^ _ / Here y = V 2 - x 2 ; and, if x, y be }!- the coordinates of the point P (fig. 7), the area Al N is represented by the interal Let x = a cos <p, and the integral transforms into ab a 53 ~r u Hence, the area of the elliptic sector APCPj is equal to ab cos- 1 a If the scctorial area APCPj^ be represented by S, the preceding result gives y . S y = sin - ; b ab where <h=^ . ab Ex. 2. The equation of a hyperbola referred to its axes is Accordingly, if x, y be the coordinates of the point P on the curve (fig. 8), the area APN is represented by b rx b , ab , x + ^/lt^^a? Consequently the area of the hyperbolic sector AGP is repre sented by . ab , This relation has given rise to a class of expressions called hyperbolic functions. Thus, if S denote the area of the hyperbolic sector APCPj, we have Hence, from the equa tion we get Let -be represented by v. and we have ab x c" +c v y c v -c~ v a = ~~ 2 ~ W = 2 In analogy with the formulae for the ellipse the expressions c" +c~ v c v - e~ v 2 ~2~~ are called the hyperbolic cosine and hyperbolic sine of v respec tively, and are usually written cosh v, sinh v ; and we have b Again, for simplicity, the hyperbola may be assumed equilateral, and a = b = l ; in this case the equations become a: = cosh v , 7/ = sinh v,
