Cylindrical coordinates where $z$ axis is not axis of symmetry. I'm a little bit uncertain of how to set up the limits of integration when the axis of symmetry of the region is not centered at $z$ (this is for cylindrical coordinates). The region is bounded by $(x-1)^2+y^2=1$ and $x^2+y^2+z^2=4$. This is my attempt:
Let $x=r\cos\theta$ and $y=r\sin\theta$.
We are bounded on $z$ by $\pm\sqrt{4-r^2}$. We take $0\leq r\leq 2\cos\theta$ and $-\pi/2\leq \theta \leq \pi/2$ to account for the fact that the cylinder is not centered with the $z$ axis (shift on the $x$ axis). The volume is given by
$$
V = \int\limits_{-\pi/2}^{\pi/2} \int\limits_0^{2\cos\theta}\int\limits_{-\sqrt{4-r^2}}^{\sqrt{4-r^2}} dz\,(r\,dr)\,d\theta \ .
$$

With all due respect, the OP's solution is correct. The circle $(x-1)^2+y^2=1$ sits inside the circle $x^2+y^2=4$, so the region in question projects onto the entire disk $\{(x-1)^2+y^2\le 1\}$. Thinking about $z$ cross-sections is awkward and to be discouraged.
