Let $ S = \{(x,y) \in \mathbb R^2 : x^2+y^2 < 1\}$ be the unit circle in $\mathbb R^2$. Let $(X_1, Y_1), (X_2, Y_2),$ be independent, both having uniform distribution over $S$. Let $D$ denote the Euclidean distance between $(X_1, Y_1)$ and $(X_2,Y_2)$. Show that $E(D^2) = 1$.
Here I integrated $$\int_0^1 \int_0^1 \int_0^{\sqrt{1-Y_2^2}}\int_0^{\sqrt{1-X_1^2}} (X_1 - X_2)^2 + (Y_1 - Y_2)^2 dX_1 dY_1 dX_2 dY_2$$