What is the easiest way to solve $\int_0^2\int_{-\sqrt{\smash[b]{1-(y-1)^2}}}^{\sqrt{\smash[b]{1-(y-1)^2}}}(2x^2+2y^2)\,\mathrm dx\mathrm dy$

Question

I am wondering if there is an easy way to calculate this integral. The original problem is to calculate the volume of the solid bouded by the cylinder $x^2+y^2=2y$, the plane $z=0$ and the paraboloid $z=2(x^2+y^2)$.

I used polar coordinates but the cylinder is deslocated.

The way I solved was to deslocate the cylinder to the origin of the space $(x,y,z)$ and therefore deslocate the paraboloid but I wonder if there an easier way that Im not seeing.