Covariance of a squared radius and its complement

Draw $X$ and $Y$ independently from $\text{Unif}(0,1)$, and set $V=X^2+Y^2$ and $U=1-X^2-Y^2$. Compute $\operatorname{Cov}(U,V)$.

$-8/45$. Since $U=1-V$, $\operatorname{Cov}(U,V)=-\operatorname{Var}(V)=-2\operatorname{Var}(X^2)$. With $E[X^2]=\tfrac13$ and $E[X^4]=\tfrac15$, $\operatorname{Var}(X^2)=\tfrac15-\tfrac19=\tfrac4{45}$, so the covariance is $-\tfrac8{45}$.