Finding the change of variables that transforms given domain into another one I was practicing some integration problem until I came upon this one. To be honest I am quite confused as to how to proceed with these question: Let find the change of variables that transforms the domain $D$ of the upper half plane whose boundary is constituted by the circles $x^2 + y^2 = 1$, $x^2 + y^2 = 4$ and the straight lines $y = 3x$, and $y = 4x$ on the rectangle $D∗ = \\{(u, v) : 1 ≤ u ≤ 4, 3 ≤ v ≤ 4\\}$ What I have tried so far its to draw both domains and see by pure logic how could I perform this change of variables, but with no luck so far I dont really know if its the right way or not. I noticed that both domains have the same area, but I dont think that helps much. Any help would be greatly appreciated. This is one of my first times use stack exchange, so Im really sorry if I did something wrong

In general, try to get some function of $x$ and $y$ equal to two different constants. Then set that function equal to $u$. Then try to find another function of $x$ and $y$ equal to two different constants and set that equal to $v$.

So your first two equations hand you $x^2+y^2 = u$ on a silver platter. The second two equations can be rewritten

$$\frac{y}{x} = 3, \frac{y}{x}=4.$$

So set $v=\frac{y}{x}.$