Let $\Omega = \left \{(x,y) : x^2-y^2 \leq 1, xy \leq 1, y \leq x, 0 \leq y \right \} \subset \mathbb{R}^2$. Find $\int_{\Omega}(x^2+y^2)dV$ using the substitution $v=x^2-y^2, u=xy$.
I'm pretty sure we can show that the substitution is a bijection (excluding $\partial \Omega) $ with $\Omega'= \left \{ (u,v) : 0 \leq u,v \leq 1 \right \}$.
And also $J^{-1}=det(\frac{\partial v,u}{\partial x,y}) = 2x^2+2y^2 \implies J = \frac{1}{2x^2+2y^2}$
But we need $J$ to be a function of $u,v$ and I don't see how we can get that, help appreciated.