Average distance between two points in a circular disk

Question

How can I find an average distance between two points lying inside a circular disk of a certain radius?

I wonder if there is any other way except of using a Monte Carlo method?

Answer 1

See the answer to this question. The expected distance is $$ d= {128 r\over 45\pi}. $$

Here is another demonstration of this result.

Answer 2

Let $D$ be the disk, and let $$M = \iint\limits_{(x_0,y_0)\in D} ~~\iint\limits_{(x,y) \in D}dxdydx_0dy_0.$$

Then the quantity you are looking for should be given by $$ \frac{1}{M}\iint\limits_{(x_0,y_0)\in D} ~~\iint\limits_{(x,y) \in D} \sqrt{(x-x_0)^2+(y-y_0)^2}dxdydx_0dy_0 $$ where a suitable change to polar coordinates in both double integrals would probably be helpful.