This problem is similar to this however, in my case my cylinder is not at (0,0).
Use double integrals to determine the volume of the solid bounded by: $$x^2+y^2\leq2y \qquad z\leq2-x^2-y^2 \qquad z=0$$
When $z=0$ we have the following plot
$$(\sqrt2)^2=x^2+y^2 \qquad x^2+(y-1)^2=1$$
Thus, the integration region is defined by the intersection. Which method should I use to integrate?