Change of Variables & the Jacobian · Challenge problem

Let $R$ be the region in the first quadrant bounded by the hyperbolas $x^2 - y^2 = 1$, $x^2 - y^2 = 4$, $xy = 1$ and $xy = 3$. Evaluate $\displaystyle\iint_R (x^2 + y^2)\,dA$.