Originally Posted by

**mr fantastic** For the spherical coordinates, obviously $\displaystyle 0 \leq r \leq 1$ and $\displaystyle 0 \leq \theta \leq 2 \pi$. So you need to get the range of the azimuthal angle.

I suggest looking at the volume side on:

The cross-section is the part of the circle $\displaystyle x^2 + z^2 = 1$ bounded by the lines $\displaystyle z = -\frac{1}{\sqrt{2}}$ and $\displaystyle z = \frac{1}{2}$.

You know the z-coordinates of the points of intersection of the lines and the circle so you can use simple trigonometry on the obvious right-triangles to get the maximum and minimum azimuthal angles.