Integral of 1/norm on a surface of a ball not centered around the origin

Question

Let $0 \neq a \in \Bbb R ^3$, and denote $S(x,r) = \{ x\text{ | } x_1^2+x_2^2+x_3^2=r^2 \}$.

Calculate $\int_{S(a,\frac{|a|}{2})} \frac{1}{|x|}dx$.

I think the divergence theorem should help here, but I'm not how. Any clues?

Answer 1

Turns it can be easily solved using the mean value theorem for Harmonic functions.

Since $\frac{1}{|x|}$ is Harmonic in $S(a,\frac{|a|}{2})$, we get:

$\frac{1}{|a|}=\frac{1}{Vol_n(S(a,\frac{|a|}{2}))} \int_{S(a,\frac{|a|}{2})} \frac{1}{|x|}dx$