Evaluate the integral F • dr along C

Question

Evaluate the integral $F • dr$ along $C$ where $F$ is the vector function $F(x,y, z) = < -y^2, x, z^2 >$ and $C$ is the curve of the intersection of the plane $y + z = 2$ and the cylinder $x^2 + y^2 =1$. Orient C counter clockwise when viewed from above.

My intuition tells me this curve $C$ is an ellipse. And I can use Stokes theorem to evaluate the integral. But if I did not have stokes theorem and I wanted to compute the integral directly, how would I parametrize the curve.

Answer 1

Hint. Use the paratmetrization $x=\cos \theta$ and $y=\sin \theta.$ Sketch a diagram.

Answer 2

Parametrize as follows: \begin{cases} x=\cos t\\ y=\sin t\\ z= 2-y = 2-\sin t \end{cases} with $0\le t \le 2\pi$. Then, $$ \int_C \vec{F}\cdot d\vec{r} = \int_0^{2\pi}\pmatrix{-\sin^2 t \\ \cos t \\(2-\sin t)^2}\cdot \pmatrix{-\sin t \\ \cos t \\ -\cos t}\; dt=\int_0^{2\pi}\sin^3t +\cos^2 t-\cos t(2-\sin t)^2 \; dt = \;... $$