To evaluate the integral
$$\iint_D \sqrt{x^2+y^2}dA$$ $$D=\{(x,y)\mid0\leq(x-1)^2+y^2\leq1 \}$$ it should be best to change variables into polar coordinates to get
$$\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\int_0^{2cos\theta} r^2drd\theta$$
Every problem like the above I have seen are done by integrating with respect to $r$ then $\theta$.
Most often these integrals are done over a circle centered at the origin so the limits are constants and the order of integration is changeable through Fubini's theorem. I have not yet seen one where the limits are functions of r and the integral is first done with respect to $\theta$.
If I decided to switch the the order of integration of the previous integral would I end up with the following integral?
$$\int_{0}^{2}\int_{-cos^{-1}\frac{r}{2}}^{cos^{-1}\frac{r}{2}} r^2 d\theta dr$$
This integral seems far worse than the previous, so I also ask how often would switching the polar bounds be beneficial? I would assume this should only be done if $\theta=f(r)$. Is that correct?
Any other information or resources on integrating with respect to $\theta$ would also be appreciated.