# Determining the limits of integration of a cardioid using a unique choice of variables

I am attempting to perform the following the double integral:

Previously, I was asked to prove the area element of polar coordinates, specifically, showing that $dA = u\ du \ dv$ for $u = \sqrt{x^2+y^2}$ and $v = \tan^{-1}(y/x)$.

And I believe it's being shoved in my face to use these variables to integrate the function I see in the figure. This is clearly polar coordinates, and I would agree this would be a good idea in this case.

Now, where I run into problems is determining the limits of integration of this function. I know the limits for $v$, which would basically be $0$ to $\pi/2$, but for $u$ it goes from $2$ to... I'm not sure.

Given the two functions, $x^2+y^2 = 4$, this is clearly $u =2$, but with the cardioid, it's $x^2+y^2=2\sqrt{x^2+y^2}+2x$, so $u^2 = 2u + 2x$? I don't know how to make this work out.

How do I work out the $u$ limits? Perhaps $2x = 2u\cos\theta$ since this is basically polar? But then I have

$$2<u<\sqrt{2u(1+\cos\theta)}$$

Which I don't think works out, because I can't have limits in terms of $u$ to work this out, as the limits either have to be in terms of other variables or constants to end up being solveable. What should I do?

$$u^2 = 2u + 2u \cos v$$
Dividing by $u$,
$$u = 2 + 2 \cos v$$
Hence the upper limit is $2+2\cos v$.