Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 5 / a

In the plane $x=0$, the right half of the cross-section has coordinates $y=a(1-\cos \theta )\sin \theta$, $z=a(1-\cos \theta )\cos \theta$, $0\le \theta \le \pi$. Reflect across the $z$ axis for the other half. The cardioid has a cusp at $(y,z)=(0,0)$ for $\theta =0$, crosses $z=0$ at $(\pm a,0)$ for $\theta =\pi /2$, and reaches $(0,-2a)$ for $\theta =\pi$. The small upper lobes reach $z=a/4$; the widest section is below the equator.