. Differential and integral calculus, an introductory course for colleges and engineering schools. of any solid. This method consists in regarding thevolume as the limit of the sum of an infinite number of rightcylinders of infinitesimal altitudes, and in calculating this limitby integration. If A (x) be the area of the base of such a cylinderat distance x from the yz-pl&ne, and if dx be its altitude, then X = b r>h V = lim Z,A(x) dx = J A{x) dx. 1. To find the volume of the ellipsoid, + y- +? l. Conceive the ellipsoid to be cut into thin slices by planes parallelto the i/z-plane. Each of t
. Differential and integral calculus, an introductory course for colleges and engineering schools. of any solid. This method consists in regarding thevolume as the limit of the sum of an infinite number of rightcylinders of infinitesimal altitudes, and in calculating this limitby integration. If A (x) be the area of the base of such a cylinderat distance x from the yz-pl&ne, and if dx be its altitude, then X = b r>h V = lim Z,A(x) dx = J A{x) dx. 1. To find the volume of the ellipsoid, + y- +? l. Conceive the ellipsoid to be cut into thin slices by planes parallelto the i/z-plane. Each of these planes cuts the-slirface in an ellipse. On each of these ellipses as baseconstruct a right cylinder whoseother base lies in the adjoiningplane. The volume of the ellip-soid is the limit of the sum ofthe volumes of these cylindersas their number increases with-out limit. This is fairly obviousfrom the figure, and the statement can be proved rigorously by amethod similar to that used in Arts. 175 and 176. We do notgive the proof. Let the bases of one such cylinder lie in the planes 286. / §192 VOLUMES OF SOLIDS 287 x = Xi and x = X\ + dxi. Setting x\ for x in the equation of theellipsoid, we get, after simple transformation, r + = 1. This is the equation of that base of our cylinder which lies in theplane x = X\. The area of this base is irbcl 1 —J, and there-fore the volume of the cylinder in question is irbcl 1 jjdxi. Dropping subscripts, we have for the volume of the ellipsoid v= ThcJ-a \l a*r = Tbclx ~ 3^J_a= rabc- The same result would of course have been obtained by cutting the ellipsoid into slices parallel to either of the other coordinate planes. Note that when two of the quantities a, b, c are equal, the foregoing formula gives the volume of the ellipsoid of revolution, and when a = b = c, the formula gives the volume of the sphere. x^ v^2. To find the volume cut from the paraboloid — + j-2 = 2 z by the plane z = mx. The solid whose volume is required is OAPBC
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