Find the volume of the solid that is generated when the region in the first quad

Question

Find the volume of the solid that is generated when the region in the first quadrant bounded by y= x^2/9 , y=1 and x=0 is revolved about the line y=-1 .

Explanation / Answer

a) Volume V= piS(x=0 to pi/2)sin^2(x)dx=.....(S=the sign of integration) (pi/2)S(x=0 to pi/2)(1-cos(2x))dx= (pi/2)(x)(x=0 to pi/2)= (pi^2)/4 b) Volume V= piS(y=0 to 1)(pi/2-arcsiny)^2(dy) Let y=sinz, then dy=(cosz)dz=> V=piS(z=0 to pi/2)[(pi/2)^2-(pi)z+z^2] cosz(dz)=> V=[(pi^3)/4]S(z=0 to pi/2)cosz(dz)- (pi^2)S(z=0 to pi/2)zcosz(dz)+ (pi)S(z=0 to pi/2)(z^2)cosz(dz)=> V=pi(pi^2+pi+2) c) The volume V= pi(2^2)(pi/2)-pi(pi/2)-piS(x=0 to pi/2)[ 1-sinx)^2]dx=> V=(3/2)pi^2-piS(x=0 to pi/2)( 1-2sinx+sin^2x)dx=> V=(3/2)pi^2-3pi^2/4+2pi=> V=(3pi^2)/4+2pi=> V=pi(3pi/4+2)

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