Let $F=(xy,z,y)$. Let $S$ be the boundary of the solid determined by
\begin{cases} x+y+z\le 18\\ x^2+y^2\le 4\\ x,y,z\ge 0 \end{cases}
I've drawn pictures, and the solid is just a quarter of a cylinder with radius $2$ cut above by a plane, so, to parametrize it, I decided to separate my surface in $3$ regions: the wall, the floor and the ceiling, and came up with this
$$ \begin{align} \Sigma (r,t)=(2\cos t, 2\sin t, r)\quad &0\le t \le \frac \pi 2\, ,0\le r\le 18-2(\cos t+\sin t) \tag 1\\ \psi(r,t)=(r\cos t, r\sin t, 0)\quad &0\le t\le \frac \pi 2\, ,0\le r \le 2 \tag 2 \end{align} $$
Where $(1)$ is the parametrization of the wall and $(2)$ is the parametrization of the floor.
I couldn't come up with a parametrization $\delta (r,t)$ of the ceiling, as it looks to be (from my pictures) a "triangle" with a curved side.
Could someone help me with that parametrization?
Having these three parametrizations, how do I compute the integral? I know how to do this for scalar fields, I just evaluate the function $f$ at the parametrization ($f(x,y,z)\mapsto f(\Sigma(r,t))$)and multiply it by $||\Sigma_r\times \Sigma_t||$ and calculate a regular double integral, but how is this done with vector fields?