I am struggling with a specific type of double intergrals. Example, Calculate $\iint_D \sqrt{4(x^2+y^2)+1}dxdy$ where D is given by $(x-1)^2+(y-1)^2\leq 4$.
I have tried to do a variable change $x=1+r\cos(\varphi),y=1+r\sin(\varphi)$ which gives me easy bounds but then I am stuck with $\iint_E r\sqrt{4(r^2+2r(\cos\varphi+\sin\varphi)+2)+1}drd\varphi$.
How should one attack a problem like this one?