Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 2 / b / Solution

The cardioid has reflection symmetry about the $x$ axis because $1+\cos (-\theta )=1+\cos \theta$. It passes through $(2,0)$ at $\theta =0$, through $(0,\pm 1)$ at $\theta =\pm \pi /2$, and has a cusp at the origin when $\theta =\pi$. These points and the following sketch show its orientation.

Using the polar coordinates formula for arc length,The absolute value changes sign at $\theta =\pi$, so