Past exam of the mathematical methods II course of the University of Oxford / 2013 / 5 / b / Solution

In polar coordinates, the area element is $rdrd\theta$. Using the bounds from part (a),Since the bracket equals $-3/4-2\cos \theta +(1/2)\cos 2\theta$, an antiderivative is $-3\theta /4-2\sin \theta +(1/4)\sin 2\theta$. Evaluation givesThe inner-circle contribution must be subtracted before angular integration; integrating the entire cardioid or the entire circle would include regions excluded by the specified intersection.