Have a look to the scheme below
means that you integrate f on the domain defined by and for each value of x
When you plot the function you can see that it is a quarter of a circle.
x between 0 and 1, and y between 0 and defines the quarter of the disk as area of integration (see red line).
If you want to change to polar co-ordinates, you have to sweep the same domain using and r.
You can see that you can manage this by choosing and for each value of
Now you have to substitute and
to get finally