Originally Posted by

**Ackbeet** Reply to AllanCuz at Post # 5:

I think you're a bit confused. The polar angle in this problem is $\displaystyle \phi,$ and inhabits the natural range $\displaystyle [0,\pi]$ in any spherical coordinates problem. The azimuthal angle in this problem is $\displaystyle \theta,$ and inhabits the natural range $\displaystyle [0,2\pi]$ in any spherical coordinates problem. However, in this problem, the shape described, which is that shape bounded by the $\displaystyle xz$ plane, and the two hemispheres $\displaystyle \rho=3$ and $\displaystyle \rho=4$, you have the artificial restriction $\displaystyle \theta\in[0,\pi].$ 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.