I am currently struggling with one question from an assignment:
Calculate the line integral
$$\int_C\ F\cdot \text{d}r ~~~~~~~~~~~ \int_C\ G\cdot \text{d}r$$
where $C: \left\{z = 4 \cap z = \frac{x^2}{4} + y^2\right\}$
in the first octant.
From here I think the upper boundary is $(0,0,4)$ and no idea about the lower one.
I know that $z=\frac{x^2}{4} +y^2$ is an ellipse where $-2\leq y \leq 2$ and $-4\leq x\leq 4$ but since it's in the first octant it's $0\leq y \leq 2$ and $0\leq x\leq 4$
Also from the questions leading up to this $F$ is not conservative and $G$ is conservative.
Any help would be appreciated, thanks.
A possible parametrization of $C$ is $$ \gamma(t) = (4\cos(t),\ 2\sin(t),\ 4), \ \ \ t \in [0,\pi/2] $$
The fact that $F$ is conservative means that $\oint F \cdot dr = 0$. In this case this means that if we were to integrate over the whole ellipse it would gives us $0$ but since we are only integrating over a quarter of it, there is no reason for it to be $0$ (even though it can).