Reply to AllanCuz at Post # 5:

I think you're a bit confused. The polar angle in this problem is

and inhabits the natural range

in any spherical coordinates problem. The azimuthal angle in this problem is

and inhabits the natural range

in any spherical coordinates problem. However, in this problem, the shape described, which is that shape bounded by the

plane, and the two hemispheres

and

, you have the artificial restriction

There is no restriction on the polar angle.

Hence your limits, while producing the same answer for the integral as my limits, do not describe the shape in the original problem.

Draw a picture (as I did); that might help you see what I'm talking about.