Question: Evaluate $∫F$.Nds where $F = 2x^2y \hat{\imath} -y^2 \hat{\imath} + 4xz^2 \hat{k}$ and $S$ is the closed surface of the region in the first ocant bounded by the cylinder $y^2+z^2 = 9$ and the planes $x= 0 ,x=2 , y=0, z =0$.
In this question how do I find the value of $ds$ for curved surface. In the book the $ds$ is given $ds= \frac{dydx}{(z/3)}$ is obtained by mod of normal vector dot unit $z$. Why did we do this? And why didn't we do this for the plane surfaces?
Also for curved surface how do I determine whether I use $dxdy$ or $dydz$ or $dzdx$?