Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 8

Reflection in the $y$-axis preserves the disk but changes the sign of the integrand $xy$. The double integral is therefore its own negative and must beEquivalently, for each fixed $y$ the integrand is an odd function of $x$ on a symmetric interval.

The curl of a gradient is zero pointwise, so its surface integral over this sphere isFor this particular polynomial, $\nabla \Phi =2(x,y,z{)}^T$, whose curl vanishes directly as well.