Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 15 / a / i / Solution

The radial interval is nonempty exactly when $1+\cos \varphi \ge 3/2$, or $\cos \varphi \ge 1/2$. Choose the representative angular range $-\pi /3\le \varphi \le \pi /3$. The region of integration is thereforeIt is the right-hand cap inside the cardioid and outside the circle of radius $3/2$, symmetric about the horizontal axis. Its boundary intersections have coordinates $(3/4,\pm 3\sqrt{3}/4)$, and the cap reaches $(2,0)$. The radial segment on the horizontal axis runs from $3/2$ to $2$.