"Use the surface integral in Stoke's Theorem to calculate the circulation of the field F=$⟨x^2,2x,z^2 ⟩$ around the curve $C,$ where $C$ is the ellipse $4x^2+y^2=4$ in the $xy$ plane, counterclockwise from above."
My Work So Far:
The circulation is $\int_C$F$\cdot dr$, meaning that it can be found with $\int\int_S curl$ F $\cdot ds$
$curl$ F $= ⟨ 0,0,2⟩$
$\int\int_S ⟨ 0,0,2⟩$ $\cdot ds = \int_C$F$\cdot dr$
From here, though, I'm confused about what to do for $ds$ and the boundaries. The boundaries in particular are strange because of the elliptical shape; polar coordinates aren't appropriate because there's no circular component, but trying to remain in $x$ and $y$ results in a very ugly $y_{-\sqrt{4-4x^2} \to \sqrt{4-4x^2}}$, which doesn't seem right either.
I feel like this is intended to be a fairly simple example of Stoke's theorem but I'm having a lot of trouble wrapping my head around it. Would anyone be able to point me in the right direction?
Hint:
You can evaluate the integral as: $$\int\int_S ⟨ 0,0,2⟩\cdot ds= 2\int_{-1}^{1} dx\int_{-2\sqrt{1-x^2}}^{2\sqrt{1-x^2}} dy=8\int_{-1}^1 \sqrt{1-x^2}dx $$
can you see that this is double the area of the ellipse of half-axis $1$ and $2$?