So in the problems I encountered, I find it confusing about the domain of $\theta$.
Problems take the form: For arbitrary function $f(x,y)$, and $$\displaystyle \iint_S f(x,y)dxdy=\iint_T f(r\cos\theta,r\sin\theta)rdrd\theta$$. Find $T$ for a given $S$.
In one question, $S=\{(x,y)|x^2+y^2\leq 2x\}$, and the answer is $$\displaystyle\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \int_0^{2\cos\theta} f(r\cos\theta,rsin\theta)rdrd\theta$$
In another question, $S=\{(x,y)|a^2\leq x^2+y^2\leq b^2\}$ for a,b positive, and the answer is $$\displaystyle\int_{0}^{2\pi}\int_{a}^{b}f(r\cos\theta,r\sin\theta)rdrd\theta$$
I understand how both are correct, my only question is, would it be correct if I write the first one as
$$\displaystyle\int_{0}^{\frac{\pi}{2}} \int_0^{2cos\theta} f(rcos\theta,rsin\theta)rdrd\theta+\int_{2\pi-\frac{\pi}{2}}^{2\pi} \int_0^{2cos\theta} f(rcos\theta,rsin\theta)rdrd\theta$$
and the second one
$$\displaystyle\int_{-\pi}^{\pi}\int_{a}^{b}f(r\cos\theta,r\sin\theta)rdrd\theta$$
I get confused because in principle $\theta=\theta-2\pi$ for $\theta\in[0,2\pi]$, but obviously as integral $\theta$ is just a dummy variable.