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Suppose that $F(x,y,z) = \langle x^2z^2,y^2z^2,xyz\rangle$. Using Stoke's Theorem: $ \iint \operatorname{curl}(F) \, dS = \int_c F \, dr$, how do you evaluate this integral where $S$ is the part of the paraboloid $z=x^2+y^2$ that lies inside the cylinder $x^2+y^2=4$, oriented upward?

I see that via substitution $z=4$. But where to go from here? What is this question really asking? Am I doing a surface integral or line integral?

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The point in these questions is usually that one of the two integrals is nasty, while the other is doable. Here it looks like the one you have to do is the line integral (disclaimer: it could be the case where you are asked to verify Stokes' Theorem, which means calculating both integrals).

So $C$ is the circle of radius $2$, centered around the $z$-axis at height $z=4$. That is, $$ C:\ (2\cos t,2\sin t,4),\ \ t\in[0,2\pi]. $$ Then $$ \int_CF\,dr=\int_0^{2\pi}F(2\cos t,2\sin t,4)\cdot(-2\sin t,2\cos t,0)\,dt =\int_0^{2\pi}(32cos^2t,32\sin^2t,16\sin t\,\cos t)\cdot(-2\sin t,2\cos t,0)\,dt =\int_0^{2\pi}(-64\cos^2t\sin t+64\sin^2t\,\cos t)\,dt=0 $$

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