Finding bounds of integration

Question

Let $S$ be the region in the first quadrant of the $xy$-plane bounded by the $x$-axis and the parabolas $$x=1-\dfrac{1}{4}y^2,$$ $$x=\dfrac{1}{4}y^2-1$$ and $$x=4-\dfrac{1}{16}y^2.$$ Use the substitution $$(x,y)=(u^2-v^2,2uv)$$ to evaluate $$\iint xy \, dx \, dy$$ over the region $S$. I know that I'm supposed to find the Jacobian (which is $4(u^2+v^2)$) and then find the bounds of integration for $u$ and $v$. I have no idea how to find them. Any hints or strategies is appreciative.