Based on Calculus II calendars everywhere, apparently polar area integrals are something we expect freshmen to fully grasp after one single lecture. (Or even less: the one single lecture is often split between polar integrals and polar derivatives.)

The idea behind polar integrals, leading to the formula $\int_{\alpha}^{\beta}\frac12 f(\theta)^2\,d\theta$, is beautiful and nice. But then we expect freshmen to have no trouble figuring out the correct $\alpha$ and $\beta$ to use. In my experience, finding the right $\alpha$ and $\beta$ in some run-of-the-mill polar area exercises is harder for many students than everything else in Calculus II, combined!

I have never, after years of teaching, been able to effectively teach how to find $\alpha$ and $\beta$. I could do an okay job teaching it if you gave me a week to devote to it, but usually only one lecture or less can be spared for it.

So how do you teach polar integrals, especially how to find the correct bounds, to freshmen in a single lecture?

  • $\begingroup$ I am confused by how you can end up in a situation where it is not immediately apparent from the context. $\alpha$ and $\beta$ are after all just dependent on how $f(\theta)$ was parametrized. Only their difference $\beta - \alpha$ really matters intuitively. Example? $\endgroup$ – Squid Mar 20 '14 at 20:56
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    $\begingroup$ @Squid: The prototypical example would be, find the area of the region bounded inside $r=\cos(\theta)$. Obviously you integrate from $0$ to $2\pi$, right? Wrong! You integrate it from $0$ to $\pi$, since $\pi$ to $2\pi$ repeats the curve a second time. Now make the curve more exotic, with lots of leaves and stuff, and the students are in Death Valley without a canteen $\endgroup$ – Anonymous Mar 20 '14 at 21:10
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    $\begingroup$ @Squid: Yes, but at big state U, I do not design the exams. The exams might say simply "Find the area of the region bounded inside $r=\cos(\theta)$." All of your remarks are perfectly fine, but how do you spread this insight to undergrads in a single lecture, given that that lecture also has to derive the area formula in the first place, cover other examples, possibly cover polar differentiation, and that the students only learned about polar coordinates -period- one lecture ago, and they don't exactly have PhDs in trigonometry? $\endgroup$ – Anonymous Mar 20 '14 at 21:28
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    $\begingroup$ @Anonymous: Well I still think the "from first zero to second zero"-principle I outlined should be perfectly graspable to your students. In the case where there is a single bounded region there should be no room for error so what remains ought to be to confront them with an example when things are not so simple. I suggest showing them something like this and asking them to compute the size of each "leaf" should instill them with the concept of different bounded regions and how to look for them. $\endgroup$ – Squid Mar 20 '14 at 21:45
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    $\begingroup$ This one is a better curve than the one I gave before as a suitably more complicated example which is easier to visualize. But really, the point is as always is to ease them towards complexity from simple cases and just giving them pictures to go along with the problems at least initially is probably a good idea. $\endgroup$ – Squid Mar 20 '14 at 22:04

When I have taught polar representations of regions I used to use a physical pointer (a meter stick when at the board, a pen or pencil when working on paper) to "sweep out" the region like the hand of a clock. For instance, in the case of $r = \cos(\theta)$, I would start by pointing from the origin along the positive x-axis. I would say: "When $\theta$ is zero, the radius is 1." As the pointer sweeps counter-clockwise I would say: "As $\theta$ increases, you can see the radius is dropping, until" (now the pointer is pointing directly upwards) "now we are at 90°, or $\pi/2$ radians, and the radius is zero." Continuing to sweep counter-clockwise: "Now $\theta$ is larger than 90° and the radius is negative. See how the points on the graph are on the ray pointing in the opposite direction from the pointer?" Continuing to sweep: "We are approaching $\theta$ = 180° or $\pi$ radians, and the radius is approaching $-1$."

End by summarizing: "The pointer had to sweep out just 180°, and the function traced out an entire circle. So we integrate $\theta$ from 0 to $\pi$."

A couple of comments on this approach:

  1. First of all you notice that I switch back and forth between degrees and radians. I think that is an important support for students at this level. Usually degrees are second nature to them, but radians are still unnatural.
  2. The reason the "sweeping out" approach is helpful is because it coordinates the visual notion of "angle" (which is hard to read directly off the graph) with the experiential notion of "time".
  3. Of course the same curve could also be parametrized over the interval $-\pi/2 \leq \theta \leq \pi/2$, which has the advantage that only positive values of $r$ are involved. You could use the "sweeping out" approach to use this parametrization, too: After reading $\pi/2$ you say something like "So we've gone from 0 to 90 degrees, and so far we have swept out the top half of the circle. Now let's reset back to 0 and rotate in the other direction. See how as we go from 0 to -90 degrees we sweep out the bottom half of the circle? So the entire circle can be swept out if we go from -90 to 90 degrees."
  4. Having said that, I think there is pedagogical value in trying to acclimate students to the idea that a point can have coordinates $(r, \theta)$ where $r$ is negative, and what that means.
  5. You can also draw several radial lines inside the bounded region as "traces" of the pointer that remain in the diagram after the moving demonstration is complete. That makes it possible for students to look back at their notes later and reconstruct the process.

In the diagram below, the blue lines represent the pointer directions; the red dotted lines represent the negative values of $r$.

enter image description here

Here it is as an animated GIF; in this one, the blue rays represent the pointer direction and the red dotted segments represent the values of $r$ (whether positive or negative).

enter image description here


A lot depends on what your students have seen earlier. It is, indeed, nearly impossible to do a good job in a single 2hr lecture, if you need to cover both differentiation and integration in polar coordinates in one go. The good ole spiral approach (bad pun intended) could work better. I describe how I do it, but unfortunately I don't know if this is any better. Alas I cannot cite any comparative studies.

Remedy: draw pictures! Lots of them! Insist that the students do so, too. It's nearly impossible to do well here if you don't.

IMO the basic problem is that students are not used to thinking in terms of polar coordinates, and polar coordinate curves, at all. In school they at least got to plot several lines, parabolas, trig functions, rational functions and such (YMMV). Thus they know what to expect when plotting $y=f(x)$ for some standard $f$. Not so in the polar world.

After having banged my head against that wall for a year two I was in a position to partially redesign the 1st year calculus courses (read: write my own lecture notes). In these parts we do elementary functions, sequences, continuity and derivatives in the Fall, and the integration and series in the Spring. Except that my predecessor had done indefinite integrals at the end of the Fall semester. I moved that to the Spring and instead did differentiation on parametrized curves, including polar ones. I only set modest goals. While I did the obligatory example of tangents on a logarithmic spiral I largely only expected students to be able to plot in polar. A typical exam question is to determine which of given six polar curves come from which functions. They do well with four-leaved clovers and such, but even good students may err in noticing the difference between a left-handed and a right-handed spiral.

Anyway, come Spring term and integration at least the good and intermediate students usually do well. Weaker brothers and sisters still think in terms of $xy$-plots out of inertia. Because they never drew those pictures.

Which brings us back to the pictures. Even relatively good students fail to draw enough of them. So the test question:

Find the area of the region bounded by the polar curves $r=\cos\varphi$ and $r=\sin\varphi$.

does not go as well as I had hoped. Way too often they only tried to solve the equation $\cos\varphi=\sin\varphi$. If only they had plotted it first and started thinking second.

Edit: Did I say that I think you need to make them plot curves in polar coordinates? Many of them! And you should give them examples showing how it is done. Also those that aren't in the textbook/lecture notes.

Caveat: I'm not positive that the idea of introducing the students to polar coordinate curves at the end of a semester long course is best. Come December some of them are exhausted by the epsilons, and may not be receptive to learn about something they have never seen earlier.


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