<s>[INST] Using Divergence theorem to calculate flux 

Let $W$ be the region bounded by the cylinder $x^2+y^2=4$, the plane $z=x+1$, and the $xy$-plane. Use the Divergence Theorem to compute the flux of $F = \langle z,x,y+z^2 \rangle$ through the boundary of $W$.

So far I've gotten to the point of computing div$(F)$ and integrating from $0$ to $x+1$ to obtain $$\iint_{D}(x+1)^2 dA.$$
My problem is finding the bounds of the domain which is the circle of radius $2$ centered at the origin. I understand I must use polar coordinates but since the circle is cut off by the line $x=-1$ I'm having trouble figuring out what the bounds for the radius should be. I think $\theta$ goes from $2\pi/3$ to $4\pi/3$ (somewhat guessing the bound for theta when the radius is cut off by the line $x = -1$)
 [/INST] If $x = r\cos\theta = -1$, then 
\begin{align*}
    r = -\frac{1}{\cos\theta} = -\sec\theta
\end{align*}
But since this means you will eventually have two compute two integrals, maybe polar coordinates is not the way to go.
\begin{align*}
    \iint_D (x+1)^2\,dA &= \int_{-1}^2 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}(x+1)^2\,dy \,dx \\
    &= 2 \int_{-1}^2 (x+1)^2\sqrt{4-x^2}\,dx \\
\end{align*}
Then substitute $x=2\sin\theta$.
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