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

A scalar potential is $U(x,y)=x^{3}y+y^4$, since $\nabla U=(3x^{2}y,x^3+4y^3)$. The fundamental theorem for line integrals therefore reduces the line integral to its endpoints. The curve starts at $(0,1)$ and ends at $(2,1)$, so