I am trying to average $r$ over the solid angle $\Omega$ in 3D. To start this I have expressed $r$ in terms of the angle $a$ and sides $x$ and $d$ in 2D with the help of the law of cosines: $r = x*cos(a) + d \sqrt{(1-x^2/d^2 Sin^2(a))}$ or $r = \sqrt{x^2 + d^2 - 2xd* Cos(a)} $. The next step that I tried to take was $\frac{1}{2\pi} \int_0^{2\pi} r da *\frac{1}{\pi}\int_0^{\pi}sin(\phi)d\phi$ to average over the solid angle $d\Omega = sin(\phi)d{\theta}d\phi$. This method gave me results in terms of elliptic integrals which is not what I was looking for. Is there a simpler (or completely different) method for doing averaging over the solid angle? Or should I reconsider my expressions for $r$? Thanks
