Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics V / 1 / a / ii / Solution

In polar coordinates, put $x=r\cos \theta$ and $y=r\sin \theta$. Differentiating along the curve givesThe cardioid has $r\ge 0$ and is traversed once counterclockwise as $\theta$ increases from $0$ to $2\pi$. Its cusp at $\theta =\pi$ causes no difficulty: split the line integral there, or exclude a small arc and take its limit. Thus signed area as a boundary integral gives