Area inside Cardioid

Theorem
Consider the cardioid $C$ embedded in a polar plane given by its polar equation:
 * $r = 2 a \paren {1 + \cos \theta}$

The area inside $C$ is $6 \pi a^2$.

Proof
Let $\AA$ denote the area inside $C$.

The boundary of $C$ is traced out where $-\pi \le \theta \le \pi$.

Thus: