(879,#17) Using spherical coordinates, evaluate

where E is the part of the unit ball x² + y² + z² <= 1 that lies in the first octant:

We have

The integral is equal to Pi/30:

lprint( int( (rho*sin(phi)*sin(t))^2 *rho^2 *sin(phi), rho) );
lprint( int( (rho*sin(phi)*sin(t))^2 *rho^2 *sin(phi), rho = 0 ..1) );
1/5*rho^5*sin(phi)^3*sin(t)^2
1/5*sin(phi)^3*sin(t)^2

lprint( int(int( (rho*sin(phi)*sin(t))^2 *rho^2 *sin(phi), rho = 0 ..1), phi) );
lprint( int(int( (rho*sin(phi)*sin(t))^2 *rho^2 *sin(phi), rho = 0 ..1), phi = 0 .. Pi/2) );
1/5*(-1/3*sin(phi)^2*cos(phi)-2/3*cos(phi))*sin(t)^2
2/15*sin(t)^2

lprint( int(int(int( (rho*sin(phi)*sin(t))^2 *rho^2 *sin(phi), rho = 0 ..1),phi = 0 .. Pi/2), t) );
lprint( int(int(int( (rho*sin(phi)*sin(t))^2 *rho^2 *sin(phi), rho = 0 ..1),phi = 0 .. Pi/2), t = 0..Pi/2) );
-1/15*cos(t)*sin(t)+1/15*t
1/30*Pi

by:sh