Past exam of the mathematical methods II course of the University of Oxford / 2013 / 3 / a / Solution

For $\mathbf{r}(t)=(t,t^2,t^3)$, the endpoints correspond to $t=0,1$ and $\dot{\mathbf{r}}=(1,2t,3t^2)$. The vector field on the path is $\mathbf{A}=(9t^2,-14t^5,20t^7)$, so the line integral isIncluding the derivative of each path coordinate is essential; merely substituting the path into the field does not supply the line element.