I want to calculate a flux of a vector field $\vec F(x,y,z)=(2xy,y,x^2)$ through the surface $S$ that is the piece of the plane $z+x=4$ inside the cylinder $x^2+y^2=4x$. I used cylindrical coordinates to parameterize the surface. Then $x \to r\cos(\theta)+2$ and $y \to r\sin(\theta)$. So I'd parameterize the surface as
$$ \vec{r}(r,\theta) = \left< r\cos(\theta)+2, r\sin(\theta), 2 - r\cos(\theta) \right> $$
This way the bounds on the integral will be $0 \leq r \leq 2$ and $0 \leq \theta \leq 2\pi$.
The normal vector $\vec{N}$ is $$ \vec{N} = \vec{r}_{r} \times \vec{r}_{\theta} = \left| \begin{pmatrix} \widehat{i} & \widehat{j} & \widehat{k} \\ \cos \theta & \sin \theta & -\cos \theta \\ -r \sin \theta & r \cos \theta & r \sin \theta \\ \end{pmatrix} \right| = \langle r, 0,r\rangle, $$
Here I want to graph this surface and also I need to know how to indicate this normal vector on the graph.
I would also like to know if what I did so far is correct.