Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 3 / b / i / Solution

Parametrize the straight path by $\mathbf{r}(t)=(t,t,t)$, $0\le t\le \pi$. Then ${\mathbf{F}}_3\cdot d\mathbf{r}=(4t^3+\sin t)dt$, so direct evaluation of the line integral givesThe fundamental theorem for line integrals supplies an independent check: this is ${\varphi }_3(\pi ,\pi ,\pi )-{\varphi }_3(0,0,0)$.