The exercise is
Evaluate the double integral of the function $f(r, \phi) = r$ in the domain limited by cardioid $r = a(1 + \cos(\phi))$ and the circle $r = a$
If $T$ is the domain, I want
$$\int \int_T r \ dr \ d\phi$$
Now, I know the basic rule about integration; if can change variables and then multiply by the determinant of the resulting jacobian.
My problems are:
1) I don't quite know how to draw the set $T$ on the $xy$ plane.
2) I don't know what change of variables I should be making. I understand that the basic idea is to transform the domain from something "fancy" to a rectangle, but I am having trouble with that
I would like a deep explaination, maybe also on the polar coordinates and how to relate them to the xy plane (apart the obvious $x = r \cos \phi$), how this can be extended to other transformations etc.
I think the question could be more clear. First, you write that you want to evaluate the integral
$$\iint\limits_{T} r \, dr \, d\theta.$$
But that is not consistent with the fact that you want to integrate $f(r,\theta)=r$ over the given domain. Reading that sentence, perhaps you want to evaluate
$$ \iint\limits_{T} r \, dA = \iint\limits_{T} r \, r \, dr \, d\theta.$$
Note that, in polar coordinates, $dA = r\,dr\,d\theta$.
Next, there might be several choices for the domain.
Assuming the red portion, we get
$$2\int _{\frac{\pi }{2}}^{\pi }\int _{a (\cos (\theta)+1)}^ar^2dr\,d\theta = \frac{22 a^3}{9}-\frac{\pi a^3}{2}.$$