Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics V / 2 / c / Solution

Take counterclockwise traversal as viewed from $z>0$. In the plane $z=0$, the work is the planar line integral of $P=2x-y$ and $Q=x+y$. Green's theorem givesThe $z$ component contributes nothing because $dz=0$. The question specifies one circuit without specifying its direction: clockwise traversal gives $\boxed{-18\pi }$.