Examples

Vector line integral

Vector line integral

Problem: Compute ${\int }_{\mathcal{C}}2ydx-3dy$, where $\mathcal{C}$ is an ellipse parametrized by $\mathbf{r}(\theta )=(5+4\cos \theta ,3+2\sin \theta )$.

Solution: First calculate that $x^{\prime }(t)=-4\sin \theta$ and $y^{\prime }(t)=2\cos \theta$. Plug everything into the integrand to obtain:

$$(5+4\cos \theta ,3+2\cos \theta )=(2(3+2\sin \theta )(-4\sin \theta )-3(2\cos \theta ))d\theta .$$

Simplify this and evaluate:

$${\int }_0^{2\pi }(2(3+2\sin \theta )(-4\sin \theta )-3(2\cos \theta ))d\theta =-{\int }_0^{2\pi }24\sin \theta +16{\sin }^2\theta +6\cos \theta =-16\pi .$$