Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/64

14 INFINITESIM For example, cubature of the ellipsoid is equivalent to the determination of the triple integral fff dxdydz , for all values of x, y, z subject to the relation Here, as in all other cases, we integrate, first, with respect to one of the variables, regarding the others as constant, and determine the limits from the given relation. Thus, integrating with respect to z, and observing that the limit- AL CALCULUS volume of the solid bounded by the surface z=f(x, y), by the plane of xy, and by the cylinder having as its base the ellipse ing values of z are c / 7$ ^7? c * / 1 -- 5- - -TT we g<- t V a 2 b* in which x, y are connected by the relation ^ + ^ <:l - This integral is easily determined by making dy b A/ 1 - ~ V Ct" then and where the limits for $ are and - . - 2> But hence cos <p d<f> , M - n dx = $wabc, as before. -a a ~ J The geometrical interpretation of each step in the preceding demonstration can be readily supplied by the reader. It may be observed that, in consequence of the symmetry of tho ellipsoid, the preceding integrations might have been limited to positive values of x, y, z, thus determining the eighth part of the entire volume. A similar remark applies to any symmetrical sur face. It will also be observed that the determination of the volume of an ellipsoid is a simple case of the theorem given in 157. Similarly the volume included within the surface is reducible to the determination of the triple integral ffj dxdydz extended to all positive values of x, y, z, subject to the condition Hence, by 157, we get V = Imnabc Thus, for instance, the volume enclosed by the surface . 4:TT(tbC In like manner the volume enclosed within the surface / r / ,/ 2 / ~ 2 (fNiN-^- 1 * 3.7.11.13 and so on. 174. From the preceding it will be apparent that every double integral may, in general, be represented by a volume. As an example, let us consider the double integral V iu.i- j;" f(x , y}dxdy . Here, since y> and < /2ax -x", and the limits of x are and In, it is readily seen that the integral represents half the _. o ~r 7 >> - a- U- a- - a, For instance, suppose the bounding surface to be the paraboloid m then the volume in question becomes 2 /4< }dxdy Til J ./u To integrate this, assume x = 2 sin 2 0, and we get - T l 175. Again, the double integral -X /-Y 77 ^ ^o* i/o when the limits X, , Y, y (> are constants, represents the volume bounded by the plane xy, the surface z=f(x,y), and the planes x = X, x = x ,y Y, y = yn- Also, in the determination of this double integral the order of integration may, in general, be changed ( 148) ; and this change in the order produces no alteration in the limits. The latter statement no longer holds when the limits of integration with respect to the first variable are functions of the second. In this latter case it is of importance to be able to determine in each case what are the new limits when the order of integration is reversed. This can generally be best effected from geometrical con siderations ; thus, for instance, in the example of the preceding article, we readily find, when the order is reversed, the new limits of x to be a + a A / 1 - ^ and ^-, and that the sub sequent limits for y arc and b. As another example, let us consider the double integral If we take on the axis of a; a portion OA = (fig. 10), and on tin axis of y, OB = Z>, and complete the rectangle OACB, it is plain from the equation that the point (x, y) is limited to the triangle OAC. Accordingly, if the order of in tegration be reversed, we must suppose the area, instead of being divided into infinitesimal strips parallel to the axis of y, to be divided into strips parallel to the axis of x. Hence, the limits for x, when y is constant, are a and -- ; and the subsequent limits for y are b and 0. Consequently, 77 ^0 ^0 f(x,y}dxdy = / I f(x,y}dxdy. o /a t> As an exemplification of the advantage of an interchange in the order of integration it will suffice to take the double integral o ^/(a-x)(x-y) Here, interchanging the order, we have by the preceding But ( 139), o >/ (rt - x)(x - y) dx Jy V(rt - x)(x - y} .-. - Trf fWij = * {/() -/(O) J- . It maybe observed that in many cases, when the order of Integra-