$S$ is a closed piece wise smooth curve by traveling in straight lines in the following order: $(0,0,0)\to (2,0,4)\to (3,2,6)\to (1,2,2)\to (0,0,0)$. (Thus $S$ is on the plane $z=2x$) Evaluate $\int_S z\cos(x)dx+x^2yzdy+yzdz$.
I think of Stoke's Theorem. I found $curl(F)=(z-x^2y,\cos(x),2xyz)$. Also the normal vector of $S$ is $\frac{1}{\sqrt5}(2,0,-1)$. Thus it turns to be $\frac{1}{\sqrt5}\int_S 2z-2x^2y-2xyzds$ However, if I parametrize $S$ to be $(u,v,2u)$. Then the boundary of $u,v$ is a big problem. I don't know how to find it correctly.