Past exam of the mathematical methods II course of the University of Oxford / 2017 resit / 5 / Solution

A vector line integral along an oriented parametrized curve $r(t)$ is $\int a(r(t))\cdot r^{\prime }(t)dt$; for a force field it is the work along that path. On the upper branch, set $y=t$, $x=t^2$, $0\le t\le 1$. ThensoThe stated $x$ endpoints do not distinguish the two branches of $y^2=x$. If the lower branch is intended, the same parametrization runs from $t=0$ to $t=-1$, giving instead $\boxed{4/3}$. The two values correspond to different terminal points, $(1,1)$ and $(1,-1)$; choosing a branch is necessary to make the question unambiguous.