. Differential and integral calculus, an introductory course for colleges and engineering schools. e we have y2 = ~i(a2-x2).a2 Then V = 7T y2dx = — ) (a2 - x2)dx = — la2x - - J-a a~ J-a a2 V 3/-a ^&74 s\ |.a&2. When b = a, F = | A-a3, the well-known formula for the volume of a sphereof radius The Surface. We have , -bx , 1 . ,2 a4-c2z2 a2-e2z2 ^ = —/ , and 1 + ?/2 = -— = — • a Va2 - x2 « (a - z2) a2 - & Also, 2/ Vl + y2 = ? V^2^^ ^f ~ eV = - (a2 - e2*2)^ a Va2 - Z2 a Therefore 2rb r+ -a ae - fi = 2tt f yVl+y*dx = — f (a2 - e2x2)*dex = — [ex Va2 - e2x2 + a2 sin-1 —l+°aeL a J-a = — [a2e Vl
. Differential and integral calculus, an introductory course for colleges and engineering schools. e we have y2 = ~i(a2-x2).a2 Then V = 7T y2dx = — ) (a2 - x2)dx = — la2x - - J-a a~ J-a a2 V 3/-a ^&74 s\ |.a&2. When b = a, F = | A-a3, the well-known formula for the volume of a sphereof radius The Surface. We have , -bx , 1 . ,2 a4-c2z2 a2-e2z2 ^ = —/ , and 1 + ?/2 = -— = — • a Va2 - x2 « (a - z2) a2 - & Also, 2/ Vl + y2 = ? V^2^^ ^f ~ eV = - (a2 - e2*2)^ a Va2 - Z2 a Therefore 2rb r+ -a ae - fi = 2tt f yVl+y*dx = — f (a2 - e2x2)*dex = — [ex Va2 - e2x2 + a2 sin-1 —l+°aeL a J-a = — [a2e Vl - e2 + a2 sin1 e]and, finally, sm—^ eWhen 6 = a, e =0, and = 1, and we have the well-known formula for the surface of the sphere of radius a, £ = 47ra2. Example 2. Find the volume and surface of the ring generated byrevolving the ellipse about a line in its plane parallel to the major axisand at a distance k from it. 258 INTEGRAL CALCULUS §176 We choose for OX the axis of revolution and for OY the minor axis ofthe curve. The equation of the curve is thenY. b2x2 + a2(y - k)2 = a?b* whence k ±-Vo2a We suppose throughout that k = b. 1st. The Volume. The required vol-ume, V, is the volume generated bythe revolution of the plane figureCABAC minus the volume generatedby the revolution of CABAC. De-X noting these two volumes by Vb andVb, we have V = Vb — yB==7r f+a(k + bVa2_J-a \ a = ttJ* f> + ^ (a2 - x2) + ™ VoJ^lf] dx,Vb> = tt f+a (k--a VgT^A* ,+a dx b2, , 0N 2 kb = *( [k2 + -0 (a2 -x2)- — VoJ^2] La2 a J Hence v = iM r+« y/fzr&te = 2Mr v^3^i + a2 sin- a ./_o a L aJ_0 2 irkb (irQ2 . ttQ2 a V 2 + 2 )- 2 7r2/cafe. This may be written V= 2irk*irab. Now ?ra6 is the area of the ellipse, and 2 wk is the length of the circumfer-ence described by its center. Therefore the ring is equal in volume to aright cylinder whose base is the ellipse, and whose altitude is the distancerevolved through by its center. The volume may also be obtained
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