Is there a simple formula for this simple question about a circle? What is the average distance of the points within a circle of radius $r$ from a point a distance $d$ from the centre of the circle (with $d>r$, though a general solution without this constraint would be nice)?
The question arose as an operational research simplification of a real problem in telecoms networks and is easy to approximate for any particular case. But several of my colleagues thought that such a simple problem should have a simple formula as the solution, but our combined brains never found one despite titanic effort. It looks like it might involve calculus.
I'm interested in both how to approach the problem, but also a final algebraic solution that could be used in a spreadsheet (that is, I don't want to have to integrate anything).

By scaling, we may assume $d=1$.  Note that 
$J(r) = \int_0^{2\pi} \sqrt{1-2r\cos(\theta) + r^2}\ d\theta$ is an elliptic integral, in Maple's notation $4 (1+r) \text{EllipticE}\left(\frac{2\sqrt{r}}{1+r}\right)$.
Now you want $\frac{1}{\pi R^2} \int_0^R r J(r)\ dr$.  It appears that for $0 < R < 1$ this
can be written using a hypergeometric function as $(1-R^2)^3 {}_2F_1(5/2,5/2; 2; R^2)$.  For $R > 1$ I think you have to take the real part of that.
