Been struggling with this problem for a while and can't seem to get the bounds right. The problem:
Evaluate the surface integral $$ \int\int_S \vec{F} \cdot d\vec{S} $$
$$\vec{F}(x,y,z) = x\vec{i} + y\vec{j} + 6\vec{k}$$
S is the boundary of the region enclosed by the cylinder
$$x^2 + z^2 = 1$$
and the planes
$$y = 0 , x + y = 4$$
I have done these problems before, and as soon as I saw the bounds I immediately thought polar coordinates, however, I am stuck on how to set up the bounds with the second plane. My parametrizations that I have tried for S are $$x = cos\theta, y=y, z=sin\theta$$ which yields the cross product of the partial derivatives $$<cos\theta, 0, sin\theta>$$ I tried plugging in bounds $$cos\theta - 4 \leq y \leq 0$$ $$0 \leq \theta \leq 2\pi$$ with the dot product of the parametrized vector field to get the integral $$\int_0^{2\pi}\int_{cos\theta - 4}^0cos^2\theta+6sin\theta dy d\theta$$ which yielded the wrong answer. I then tried without switching to polar coordinates so that I could have my y values as the bounds, but I couldn't even set up that integral properly. Any help is appreciated, and any tips on writing better questions are appreciated since this is my first question.