Compute for Cov(X,Y) and Correlation(X,Y)

Question

Let $(X, Y)$ be uniform on the half disc $D = \{(x, y) : 0 < y, x2 + y2 < 1\}$.

How should I approach this problem. Should I solve double integral with inside goes from $-\sqrt1-x^2$ to $\sqrt1-x^2$ And outside goes from $-1$ to $1$?

Answer 1

Since $(-X,Y)$ and $(X,Y)$ are identically distributed, $\mathrm{Cov}(X,Y)=\mathrm{Cov}(-X,Y)=-\mathrm{Cov}(X,Y)$ hence $\mathrm{Cov}(X,Y)=0$. Likewise, $\mathrm{Corr}(X,Y)=0$.