# Change of order in polar coordinates

Evaluate $$\int\int_D \sqrt {x^2+y^2}dxdy~~~~D=\{(x,y):x\leq x^2+y^2\leq 2x\}$$ After applying the change of variables i got the integral set up as $$2\int_0^{\pi/2}\int_{\cos \theta}^{2cos \theta}r^2dr d\theta$$ But i want to change the order of integration in the polar coordinates, so i got the integral set up as $$2\left (\int_0^1\int_{\arccos r}^{\arccos r/2}r^2 d\theta dr +\int_1^2\int_{0}^{\arccos r/2}r^2 d\theta dr \right)$$ Is this correct

RegionPlot[{x<x^2+y^2<2x && y<0,Sqrt[x^2+y^2]<1 && ArcCos[Sqrt[x^2+y^2]]<ArcTan[x,y]<ArcCos[Sqrt[x^2+y^2]/2],1<Sqrt[x^2 + y^2]<2 && 0<ArcTan[x,y]<ArcCos[Sqrt[x^2+y^2]/2]},{x,0,2},{y,-1,1}]