(878,#3) Sketch the solid whose volume is given by the integral

and evaluate the integral.

Solution: The domain is that part of a sphere of radius 1, centered at the origin, in the first octant:

The integral is equal to Pi/6:

lprint( int(rho^2 * sin(phi), rho));
lprint( int(rho^2 * sin(phi), rho = 0..1));
1/3*rho^3*sin(phi)
1/3*sin(phi)

lprint( int(int(rho^2 * sin(phi), rho= 0..1), phi));

lprint( int(int(rho^2 * sin(phi), rho = 0..1), phi = 0 ..Pi/2));
-1/3*cos(phi)
1/3

lprint( int(int(int(rho^2 * sin(phi), rho = 0..1), phi = 0 ..Pi/2),t) );
lprint( int(int(int(rho^2 * sin(phi), rho = 0..1), phi = 0 ..Pi/2),t = 0 ..Pi/2));
1/3*t
1/6*Pi

by:sh