Find the change of coordinate $(x,y) = T (u,v)$ under which $$R = \left\{ (x,y) \mid 0 \le x \leq 2, x^2 \le y \le (x-2)^2 \right\}$$ is mapped to $$R' = \left\{ (u,v) \mid 0\le u \le 1, 0 \le v \le 1 \right\}$$
I am working on exercises of surface integrals. I got stuck on this problem. Could somebody please help me? Is there any general rule for that?
The (neither open nor closed) curved triangle $$\begin{align}S:&=\{(x,y)\in\Bbb R^2\mid0\le y-x^2<(x-2)^2-x^2,0\le x\le 2\}\\&=\{(x,y)\in\Bbb R^2\mid0\le y-x^2<4(1-x),x\ge0\}\end{align}$$ is mapped to the (neither open nor closed) square $$S':=\{(u,v)\in\Bbb R^2\mid0\le u<1,0\le v<1\}$$ by $$u=x,\quad v=\frac{y-x^2}{4(1-x)}.$$ The inverse bijection is given by $$T(u,v)=\left(u,u^2+4(1-u)v\right).$$