Solution (source code)

= Solution

In <spherical coordinates>, $\theta$ is the polar angle from the positive $z$ axis and $\phi$ the azimuthal angle. The <Jacobian determinant> of the coordinate transformation gives the <volume element>
$$
\boxed{dV=r^2\sin\theta\,dr\,d\theta\,d\phi},\qquad
0\leq\theta\leq\pi,\quad0\leq\phi<2\pi.
$$