How do I find the integral bounds and Jacobian to integrate this region?

Question

I just self-learned the basics of using a Jacobian transformation on a double integral of a region to make solving it much simpler. I gave myself a sample problem I pulled arbitrarily, and I'm immediately very stuck.

My understanding is at least I should let $u = x + y$, because $x + y$ ranges from 2 to 5 in this region (x + y = 2 occurring at the intersection of the 2 other functions). Regarding whether or not this first step is even correct, or what other relationship I should use to let my other variable equal something, I'm very very lost now. Can someone help? Is it even possible to use a Jacobian for this double integral? Thank you for your time. (Note: I understand there are more roundabout ways of solving this with single-variable calculus, but I intend to understand Jacobians, not just solve this problem. Thank you)

Answer 1

You could take $u=x+y$ and $v=y-x$, which up to a scaling factor just means that you're rotating the coordinate system by 45 degrees, so that the Jacobian determinant will just be a constant. Then the boundary curve $y=x^2$ becomes $\frac{u+v}{2} = \left( \frac{u-v}{2} \right)^2$, which you can rewrite on the form $v=f(u)$ by solving a quadratic equation (choosing the correct sign in front of the square root), and similarly for the other curve $x=y^2$.

Answer 2

This will get you started. Take \begin{align*} u&=\frac{\log y}{\log x} \\ v&=x+y. \end{align*} But I think inverting the Jacobian and doing the integral are quite hopeless.

Just as most single integrals cannot be integrated in elementary terms, double integrals are even less likely to be doable if you make them up too creatively.