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