Suppose we have a unit circle with center $p_0$ (let's call that circle $C_0$, this is the red circle). In addition, we have another unit circle with center $p_1$ (let's call that circle $C_1$, this is the blue circle). The distance between the two centers is $d$.
I wonder, how can I calculate the average distance from $p_0$ to the points $p$ s.t $p \in C_1 \text{ and } p \notin C_0$.
Without loss of generality, I'm trying to calculate the average distance between the origin and the points in the yellow area.
Currently, I tried to use the Law of cosines with Conditional expectation. But I don't get any insights about the problem in this way.
Thanks a lot!