# Using Divergence theorem to calculate flux

Let $$WW$$ be the region bounded by the cylinder $$x2+y2=4x^2+y^2=4$$, the plane $$z=x+1z=x+1$$, and the $$xyxy$$-plane. Use the Divergence Theorem to compute the flux of $$F=⟨z,x,y+z2⟩F = \langle z,x,y+z^2 \rangle$$ through the boundary of $$WW$$.

So far I’ve gotten to the point of computing div$$(F)(F)$$ and integrating from $$00$$ to $$x+1x+1$$ to obtain $$∬\iint_{D}(x+1)^2 dA.$$

My problem is finding the bounds of the domain which is the circle of radius $$22$$ centered at the origin. I understand I must use polar coordinates but since the circle is cut off by the line $$x=-1x=-1$$ I’m having trouble figuring out what the bounds for the radius should be. I think $$\theta\theta$$ goes from $$2\pi/32\pi/3$$ to $$4\pi/34\pi/3$$ (somewhat guessing the bound for theta when the radius is cut off by the line $$x = -1x = -1$$)

where $W$ is the region bounded by $x^2+y^2=4$, $z=x+1$ and $z=0$, i.e.,