In polar/cylindrical coordinates, does $r dr d\theta=r d\theta dr$? In polar/cylindrical coordinates, does $r dr d\theta=r d\theta dr$? For example, I believe the integral for the area of a half-circle is given by
$$\int_0^1\int_0^{\pi}{rd\theta dr}.$$
What would this integral be if the order were reversed? I find it hard to visualize taking $dr$ first.

The order of integration is not important when the other integration variables do not appear in the limits, so usually you can rearrange the integrals to your liking.  If, however, a function of $\theta$ appeared in the $r$-integral's limits, you would be forced to do the $r$ integral first.
I'm not sure why you find it hard to do the $r$ integral first, though.  Doing the $\theta$ integral first is like tracing out the whole circle at a fixed radius and then integrating over all radii.  Doing the $r$ integral first just traces out a straight line that you then integrate around a full circle.
