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

The cardioid has nonnegative polar radius $r=1-\cos \theta$, ranging from zero at the right-hand cusp to two on the negative $x$ axis. Its intersections with the circle $r=3/2$ satisfy $\cos \theta =-1/2$, giving $\theta =2\pi /3,4\pi /3$. The selected region is the left-hand cap between the two curves:The endpoints in Cartesian coordinates are $(-3/4,\pm 3\sqrt{3}/4)$.

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.