I'm looking for a solution verification:
Let $S$ be the lateral surface of the cylinder $x^2 + y^2 = 1$ bounded below by the $xy$-plane and above by the plane $y+z = 2$.
Is this a correct parametrization for $S$: $r(\theta, z) = (\cos(\theta), \sin(\theta), z)$ where $\theta \in [0, 2\pi]$ and $0 \leq z \leq 2 - \sin(\theta)$?
Then I would like to find $\iint_S F\cdot dS$ where $F(x, y, z) = [x, y, z]$ and the surface is oriented outwards.
So $r_\theta \times r_z = (-\sin\theta, \cos\theta, 0) \times(0, 0, 1)= (\cos\theta, \sin\theta, 0)$ which is already pointing outwards unless I'm wrong, so then $\iint_S F\cdot dS = \int_0^{2\pi} \int_0^{2-\sin(\theta)} (\cos \theta, \sin\theta, z) \cdot (\cos\theta, \sin\theta, 0)\,dzd\theta = \cdots = 4\pi$.