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
2026-03-28 16:28:25.1774715305
Change of order in polar coordinates
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Looks good to me. Here's a plot of your two regions of integration (top half) compared to the original region of integration (bottom half). You can see they match up perfectly.
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