Let $Q$ be the solid delimitated by the paraboloid $z = x^2 + y^2$, by the cylinder $x^2 + y^2 = 4x$ and by the plane $z = 0$. In cartesian coordinates, we can write: $$Q = \{ (x, y, z) \in \mathbb{R}^3 \ \vert \ x^2 + y^2 \leq 4x \text{ and } 0 \leq z \leq x^2 + y^2\}$$
I thought it was immediate that the solid in cylindrical coordinates could be written as: $$Q=\{(r,\theta,z)\in\mathbb R^3:0\le\theta\le2\pi,0\le r\le4\cos\theta,0\le z\le r^2\}$$ But my text says this is wrong. Why? What would be the correct description?
If we rewrite the cylinder's equation as $$(x-2)^2+y^2=4$$ we see that a ray in the $xy$-plane from the origin only hits the cylinder at $-\pi/2\le\theta\le\pi/2$. This is thus the correct bound on $\theta$.
Your bounds for $r$ and $z$ are correct.