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.
 A: 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.
A: The order does not matter except when you evaluate it by multiple integrals you are supposed to be evaluating the inner $dx_{i}$ first, then the second integral indexed by $dx_{j}$, etc. 
So for your example, the area of the half circle is the same as $$\int_{0}^{\pi}\left(\int_{0}^{R}rdr\right)d\theta=\int^{\pi}_{0}\frac{1}{2}R^{2}\theta d\theta=\frac{\pi}{2}R^{2}$$
Or $$\int^{R}_{0}\left(\int^{\pi}_{0}d\theta\right)rdr=\int^{R}_{2}\pi rdr=\frac{\pi}{2}R^{2}$$
