. An elementary treatise on the differential and integral calculus. Fig. 57. r 2a — x (3« _ x) (2ax — x*H -a (2a — x)2 hence, between the limits x — 0 and x = 2a, we haveV = 2n / (2a — xfdy = 2t / (3a—a:) (2«z—z2)* oar_ /*2a 6 a2z — 6ax* + r?;3 0 V2 o# = 2tt2«3 oa; — ^ (by Ex. 6, Art. 151). 207. Volume of Solids bounded by any CurvedSurface.—Let (x, y, z) and (x -f- dx, y -f- o#, ^ + oz) be twoconsecutive points E and Fwithin the space whose volumejs to be found. Through Epass three planes parallel tothe co-ordinate planes xy, yz,and zx; also through F passthree planes parallel to thefirst. Th
. An elementary treatise on the differential and integral calculus. Fig. 57. r 2a — x (3« _ x) (2ax — x*H -a (2a — x)2 hence, between the limits x — 0 and x = 2a, we haveV = 2n / (2a — xfdy = 2t / (3a—a:) (2«z—z2)* oar_ /*2a 6 a2z — 6ax* + r?;3 0 V2 o# = 2tt2«3 oa; — ^ (by Ex. 6, Art. 151). 207. Volume of Solids bounded by any CurvedSurface.—Let (x, y, z) and (x -f- dx, y -f- o#, ^ + oz) be twoconsecutive points E and Fwithin the space whose volumejs to be found. Through Epass three planes parallel tothe co-ordinate planes xy, yz,and zx; also through F passthree planes parallel to thefirst. The solid included bythese six planes is an infinitesi-mal rectangular parallelopipe-don, of which E and F are two opposite angles, and thevolume is dxdy dz ; the aggregate of all these solids between. Fig. 58. 384: TRIPLE INTEGRATION. the limits assigned by the problem is the required , if V denote the required volume, we have V = I J J dx dy dz, the integral being taken between proper limits. In considering the effects of these successive integrations,let us suppose that we want the volume in Fig. 58 containedwithin the three co-ordinate planes. The effect of the ^-integration, x and y remaining con-stant, is the determination of the volume of an infinitesimalprismatic column, whose base is dxdy, and whose altitudeis given by the equations of the bounding surfaces ; thus, inFig. 58, if the equation of the surface is z=f(x,y), thelimits of the ^-integration are/(£, y) and 0, and the volumeof the prismatic column whose height is Pp is f(x, y) dx dy;hence the integral expressing the volume is now a doubleintegral and of the form V = J Jf(x,y) dxdy. If we now integrate with respect to y, x remaining con-stant, we sum up the prismatic columns which form theelemental slice
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