Let $(x,y)=G(u,v)=(u^2-v^2, 2uv)$, Consider the region $A$ defined by $$u^2+v^2\le1\text{ and } u\ge0, \text{ }v\ge0.$$
let $$f(x,y)=\frac{1}{\sqrt{x^2+y^2}}$$ find the integral of $f$ over the region $G(A)$, the problem is that I'm kinda confused with the variables, let's start with the region $A$, I suspect that $G(A)$ is defined by $$(u^2-v^2)+4u^2v^2=(u^2+v^2)^2\le 1\implies (u^2+v^2)\le 1$$ and $$u^2\ge v^2, \text{ }uv\ge 0$$ Hence $$\iint_\limits{G(A)}\frac{\,dy\,dx}{\sqrt{x^2+y^2}}=4\iint_\limits{A}{\,du\,dv}$$ the last integrand is form $(f○G)(u,v)$ times the jacobian determinant, but now how can I construct the bounds of integration?