How to tell periodicity of polar curve?

Question

For example, we want to find arc length of $r=\cos(\theta)$. Students often integrate from $0$ to $2\pi$ which is wrong since the curve starts to repeat it self after $\pi$. Drawing such curve will be one way. But I can imagine much more complicated curve that is hard to draw. Is there anyway to quickly tell what the periodicity is for a polar curve.

Answer 1

Some of the comments are very focused on the technical definition of arc length and of functions, which I think is above the grasp of the student level we are working with here. We simply want to get a geometric intuition. When does the curve start to repeat itself? After that, we don't count any more arc length.

One way to help figure out when a curve begins to repeat would be to ask students to find when $r$ comes back to its initial value. However, the nuance is that in polar coordinates $r$ can be negative, so one will have to find when $|r|$ returns to its initial value. Any time a curve repeats this is a necessary condition. It is not a sufficient condition, but checking for the periodicity of $|r|$ should narrow down the set of periodicities one could check to only a few. Then you can simply find the smallest one, and integrate over that period.