Vector Calculus · Superchallenge problem

Let $D$ be the region enclosed by the cardioid $r = 1 + \cos\theta$. Using Green's theorem in the form $\displaystyle\iint_D x\,dA = \oint_{\partial D}\dfrac{x^2}{2}\,dy$, find the $x$-coordinate $\bar x$ of the centroid of $D$.