(879,#15) Using spherical coordinates, evaluate

where B is the unit ball x² + y² + z² <= 1.

Solution: We have

The value of the integral is (4/5)Pi:

lprint( int(rho^4 * sin(phi),rho));
lprint( int(rho^4 * sin(phi),rho = 0 ..1));
1/5*rho^5*sin(phi)
1/5*sin(phi)

lprint( int(int(rho^4 * sin(phi),rho = 0 ..1), phi));
lprint( int(int(rho^4 * sin(phi),rho = 0 ..1), phi = 0..Pi));
-1/5*cos(phi)
2/5

lprint( int(int(int(rho^4 * sin(phi),rho = 0 ..1), phi = 0..Pi), t));
lprint( int(int(int(rho^4 * sin(phi),rho = 0 ..1), phi = 0..Pi), t = 0..2*Pi));
2/5*t
4/5*Pi

by:sh