The potential on the shell of a sphere $|r|=R$ is given by $$\phi=\dfrac{\sin \varphi}{7+3\cos^5\theta}$$ What is the potential at the origin?
Attempted solution
The mean value theorem for potentials give $\phi(P) = \dfrac{1}{4\pi R^2} \iint_S \phi dS$ where $P$ is the center of a sphere given by $S:|r-P|=R$. This gives, with $P$ as origin: $\phi(\mathbf{0}) = \dfrac{1}{4\pi R^2} \iint \dfrac{\sin \varphi}{7+3\cos^5\theta} R^2 \sin \theta \: d\theta d\varphi = 0$ since we're integrating $\sin \varphi$ over $\varphi : 0\to 2\pi$. Does this look correct, and is there another way to solve this problem?