(879,#19) Use spherical coordinates to evaluate

where E is bounded below by the cone phi = Pi/6 and above by the sphere rho = 2.

Solution:

We have

Integrating:

lprint( int(rho^3 * sin(phi), rho ));
lprint( int(rho^3 * sin(phi), rho = 0..2));
1/4*rho^4*sin(phi)
4*sin(phi)

lprint( int(int(rho^3 * sin(phi), rho = 0..2), phi));
lprint( int(int(rho^3 * sin(phi), rho = 0..2), phi = 0 ..Pi/6));
-4*cos(phi)
-2*3^(1/2)+4

lprint( int(int(int(rho^3 * sin(phi), rho = 0..2), phi = 0 ..Pi/6), t));
lprint( int(int(int(rho^3 * sin(phi), rho = 0..2), phi = 0 ..Pi/6), t = 0..2*Pi));
(-2*3^(1/2)+4)*t
-4*Pi*3^(1/2)+8*Pi