Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 13 / d / Solution

The original PDF has angle $\theta (t)=2\pi t^n$, with $n$ an exponent on $t$. It is not $2\pi nt$: each permitted value of $n$ changes the speed but traverses the same circle once counterclockwise. Along the path, $r^2=2$ andThe radial contribution to the line integral vanishes, since $(\cos \theta ,\sin \theta ,1)\cdot (-\sin \theta ,\cos \theta ,0)=0$. The rotational contribution givesThereforeThe answer is independent of the monotone parametrization, as an oriented line integral should be.