"For example, consider integrals over all space in polar coordinates:
\[ \int_0^{2\pi}\int_0^\infty f(r, \theta) r dr d\theta \]
The upper limit of the $\theta$ integration is always 2$\pi$."
This statement is false. If $r = r(\theta)$, then it can be the case (e.g., for a "looped" limacon $r = \cos(\theta)$) that $\theta$ will not range from $[0,2\pi]$. The upper limit is not necessarily "always" $2\pi$.
\[ \int_0^{2\pi}\int_0^\infty f(r, \theta) r dr d\theta \]
The upper limit of the $\theta$ integration is always 2$\pi$."
This statement is false. If $r = r(\theta)$, then it can be the case (e.g., for a "looped" limacon $r = \cos(\theta)$) that $\theta$ will not range from $[0,2\pi]$. The upper limit is not necessarily "always" $2\pi$.