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

Let $D$ be a bounded planar region with a piecewise smooth, simple closed boundary $C$, oriented counterclockwise, and let $P,Q$ have continuous first partial derivatives on an open neighbourhood of its closure. Green's theorem states thatFor a region with holes, orient every boundary component with $D$ on its left: the outer component is counterclockwise and the inner components are clockwise.

Choose $P=-y/2$ and $Q=x/2$. Their indicated partial derivatives differ by one, so the area isThis is signed area as a boundary integral; reversing the direction of $C$ changes the sign of the integral.

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