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Evaluate the line integral $$I = \int_C F \cdot ds$$ when $F(x,y,z) = \langle x+2z, 0, y \rangle$ and $C$ is the curve of intersection of the cone $z^2 = x^2+y^2$ with the plane $2z = -x+2$.

So I am trying to use Stokes' Theorem here and can compute $curl(F) = \vec{i} + 2\vec{j}$. It's parametrizing the intersection that I'm having trouble with. I am trying to do $z = -\frac{x}{2}+1$ and then letting my curve be $$u(r,t) = \langle r\cos t, r\sin t, -\frac{r}{2}\cos(t)+1 \rangle $$ and then computing $u_r \times u_t$. But this parametrization seems off to me. Any thoughts would be appreciated!

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I have given the solution in the handwritten image. Hopefully it is clear and legible

enter image description here

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  • $\begingroup$ Im getting 8pi/3root(3) uisng Stoke's theorem $\endgroup$
    – Dom Jo
    Jul 5, 2020 at 15:13

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