# Using Stoke's theorem to evaluate an integral

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?

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 $$\int_CF\,dr=\int_0^{2\pi}F(2\cos t,2\sin t,4)\cdot(-2\sin t,2\cos t,0)\,dt =\int_0^{2\pi}(32cos^2t,32\sin^2t,16\sin t\,\cos t)\cdot(-2\sin t,2\cos t,0)\,dt =\int_0^{2\pi}(-64\cos^2t\sin t+64\sin^2t\,\cos t)\,dt=0$$