How to integrate $\iint_D 4(x^2 + y^2) \, dxdy$ on $D = \{ |z-1| \leq 1\}$.

Question

How to integrate $\iint_D 4(x^2 + y^2) \, dxdy$ on $D = \{ |z-1| \leq 1\}$. Seems like using polar coordinates is an option but what would be the boundary for a shifted circle?

Answer 1

Put $x=1+r\cos (\theta)$, $y=r\sin (\theta)$. You get $4\int_0^{2\pi} \int_0^{1} [(1+r^{2}+2r\cos (\theta)] rdrd \theta$. It is easy to evaluate this.