17
$\begingroup$

I will be teaching Calculus 1 soon, using Stewart's Calculus: Early Transcendentals as a reference. I can't help but recall my time in high school AP Calculus and my first semester undergraduate calculus, and how much I (along with many other students) struggled with the topic of solids of revolution and using integrals to compute their volumes. My personal frustration: Are all these fancy tricks and methods of finding these volumes actually useful? I never got a satisfactory answer to that question through the remainder of my undergrad years, and to this day it seems to be little more than an "I teach it because it's the standard" kind of topic. Now that I'm going to be teaching undergraduate calculus myself, I have half a mind to omit the topic altogether and spend more time on topics that might be more useful to engineering and science students, such as basic numerical methods for approximating derivatives and integrals.

Would I be doing my students a disservice if I skipped this topic altogether? Or am I right in thinking that the precious little time we have in a semester is better spent on other, more relevant topics?

$\endgroup$
12
  • 8
    $\begingroup$ I would have thought that the skills learned from working with volumes of revolution would be much more useful to engineering majors than (likely "cookbook") approximation methods. You might want to look through some standard (U.S.) 1st/2nd year engineering statics and dynamics texts in your university's bookstore (volumes of revolution ideas are scattered throughout) or visit some of the engineering faculty and talk to them. I suspect learning how to set up definite integrals for various purposes is much more important for those students than methods CAS's can/will-soon take care of. $\endgroup$ Feb 8 at 17:39
  • 17
    $\begingroup$ I'm wondering if you're letting a bad experience back when you were a student color your perceptions. $\endgroup$
    – Sue VanHattum
    Feb 8 at 17:51
  • 5
    $\begingroup$ According to the AP Calculus Standards, solids of revolution are an "AB" topic. Many people get the impression that "AB topic = Calc 1" and "BC topic = Calc 2," but that might not be the case. When I adjuncted at a community college to teach Calc I, I did not have to do solids of revolution. The layout for me was Limits, Derivatives, applications of derivatives, introduction to integrals up to u-substitution and integrals as "net change." I got the impression "applications of integrals" (finding volumes) was where Calc II started. $\endgroup$
    – ruferd
    Feb 8 at 19:22
  • 4
    $\begingroup$ Thank you all for the quick and insightful advice! I will certainly be re-evaluating my position on this topic. It's entirely possible that, like Sue VanHattum said, that I've let a previous experience sour my taste for the subject. And thanks for all the wonderful examples and resources! $\endgroup$ Feb 8 at 21:07
  • 16
    $\begingroup$ I loved it when we covered solid of revolution when I took calculus. It made all the abstract theoretical concepts real. It was an actual practical application. One of our homework assignments was to find objects around the house and figure out their volume using calculus. I actually had fun with it, and over 20 years later I still remember doing that assignment. $\endgroup$
    – Seth R
    Feb 9 at 14:40

6 Answers 6

35
$\begingroup$

An operation is born when we recognize the regularity in repeated reasoning.

Take multiplication for example. If we are living lives which involve even a modest amount of arithmetic thinking, we will encounter lots of situations where we need to calculate the total number of units present in some number of equal sized groups. We might solve these problems in ad-hoc ways at first, but it becomes more and more convenient to "formalize" the process as a new abstract operation. We can then develop algorithms for computing the product, etc.

It is amazing that one operation allows us to solve all of the following problems:

  1. I buy 34 valentines day cards (one for each of my classmates). Each costs $2.45. How much money did I spend?

  2. A cake recipe requires $2\frac{1}{3}$ sticks of butter. I want to make $2.5$ recipes. How many sticks of butter do I need?

  3. I am riding my bike at $11$ miles per hour for 2.3 hours. How many miles did I ride?

The same is true for integration. Before we "invent" the definite integral, we should understand what kinds of situations it is abstracting. Here are some:

  1. I am riding my bike for 3 hours, but my speed is variable. My speed can be described as $f(t)$ miles per hour where $t$ is the time in hours. How many miles did I travel?

  2. I am trying to find the area under a curve from $x=0$ to $x=3$, but the height of the curve is variable. The height of the curve is given by $f(x)$ inches, where $x$ is the number of inches to the right of the origin.

  3. I am trying to find the work done by raising a bucket out of a well, but the bucket is leaking so its weight is variable.

  4. I am trying to find the volume of a solid, but the area of the cross sections is variable.

In each situation the "ad hoc" method to solve the problem has a commonality: we chop the problem up into tiny pieces, approximate the answer for each tiny piece by assuming the variable quantity is "essentially constant" over the tiny pieces, and sum up the approximations. To make the approximation better, make your pieces smaller (so that the "basically constant" approximation of each piece becomes more accurate).

This process of chopping, approximating, and summing, and then taking a limit happens so frequently and in so many diverse situations that it makes sense to "formalize" this process as a new operation called integration.

Teaching the operation without also teaching what kinds of problems it solves would be similar to teaching someone how to operate a circular saw without ever showing them a piece of wood.

The volume of solids (and solids of revolution) are one example of the "integration idea": chop up the solid into lots of thin slabs, and approximate that the cross sectional area is constant so that your slab is a cylinder (whose volume you can calculate). Then sum up all of the volumes of the thin slabs. Making the slabs thinner and thinner should give better and better approximations.

While you can convey these ideas without using this particular example (there are millions of applications to choose from), volume is a classic. In particular seeing where the formulas for the volumes of cones and spheres comes from is culturally important.

$\endgroup$
7
  • 37
    $\begingroup$ Let me make explicit something that's implicit in this answer. We teach volumes of solids of revolution for the purpose of helping students learn what integration is about, not for the purpose of students knowing how to calculate volumes of solids of revolution. This should affect how we teach the topic. In particular, a student who has only memorized a formula for doing these problems has missed the point. $\endgroup$ Feb 8 at 21:54
  • 24
    $\begingroup$ Exactly @AlexanderWoo , and this also explains why it is so bad to teach this topic with "well there are 6 possibilities, so here are the 6 formulas..." which is how many textbooks approach it (and how I was taught). $\endgroup$
    – Chris Cunningham
    Feb 8 at 22:46
  • 3
    $\begingroup$ This. When I read the question, this answer was my immediate reaction. I'm not teaching students how to solve particular problems (whose solutions have been known for centuries, and which have computer implementations which are faster and more accurate than anything a human can do). Rather, I'm teaching them how to think through problems, and giving them useful general tools to help with this process. $\endgroup$ Feb 9 at 17:56
  • 1
    $\begingroup$ @ChrisCunningham I struggled with math classes taught that way. Memorizing abstract formulas and techniques is hard and kind of boring. Show me how it's actually used and it is a lot more interesting. Even if it isn't something I would personally ever do, just knowing there is a practical application always got me a lot more engaged. $\endgroup$
    – Seth R
    Feb 9 at 18:51
  • 2
    $\begingroup$ @SethR I had an interaction with a precalculus student this week very much on that topic. They wanted to know what trigonometric functions (particularly tangents) are good for. I gave a couple of off-the-cuff answers (tangents show up, for example, in aeronautical navigation when computing rate-of-climb, sines and cosines model vibrating strings, etc). I then directed her to a 3blue1brown video on Fourier analysis. The actual topic is likely way over her head, but she seemed excited to know that there were applications. $\endgroup$ Feb 10 at 17:05
13
$\begingroup$

Pardon me ignoring your Calculus question, but there is some beautiful mathematics here, e.g., Cavalieri’s principle. So there is an opportunity to connect the calculus to these "fascinating results" (to quote Xander Henderson).

Bracelet

Tom M. Apostol and Mamikon A. Mnatsakan. "Volumes of Solids Swept Tangentially Around Cylinder." Forum Geometricorum. Volume 15 (2015) 13–44. PDF download.

$\endgroup$
1
  • 4
    $\begingroup$ +1 for noting Cavalieri’s principal. I usually take a day to talk about that every time I teach the class. $\endgroup$ Feb 8 at 20:29
11
$\begingroup$

At my institution, we teach out of Thomas' Calculus (not the early transcendentals version, thank goodness). Volumes and surfaces of revolution show up in a chapter titled "Applications of Definite Integrals", which also includes arc length, some elementary fluids problems, and moments & centers of mass. My impression is that the goal of the chapter is to give students a number of examples in which integrals give geometrically or physically useful / meaningful results.

At my institution, this chapter is usually taught during the last two or three weeks of the semester. Philosophically, I think that is the right place for it. I don't expect students to master the topics, but I do think that they benefit from exposure. There are ideas in that chapter which recur, and there are some interesting problems which are resolved using elementary calculus techniques:

  • computing volumes using cross-sections foreshadows the Fubini-Tonelli theorem, which comes back in multivariable calculus;

  • computing volumes and surface areas of revolution exposes students to the idea of exploiting symmetries in order to simplify computations;

  • using these techniques, it is possible to show that a cylinder is precisely a ball plus a cone---this is a fascinating result, if you ask me; enter image description here

  • computing moments and centers of mass gives insight into a physically meaningful process (one can "numerically integrate" by plotting a curve on a material of uniform density, cutting out the region, and balancing it on a fulcrum), but also exposes students to one of the essential ideas of probability theory;

  • and so on...

I don't think that it is the end of the world if these examples are skipped, and I generally don't cover ever section in that chapter (because time is limited), but I do think that students benefit from seeing the material. It gives them something more concrete to work with, and provides a number of examples which, in some sense, justify the previous 14 weeks of lecture.

$\endgroup$
11
$\begingroup$

A point to consider that has not been emphasized much in other answers so far: removing a topic from a syllabus in a service course should not be done before getting input from instructors of other courses that have yours as a prerequisite. People who teach Calculus III might assume their students have experience computing volumes by the method you want to skip. They would not be happy to find out their students never saw the method in Calculus II first (a simpler setting). Talk not just to the math department but other departments that have their majors take your course where you think volume integrals might matter: physics, engineering, maybe not economics.

I will share a personal experience. Several years ago (well, more than just several), a colleague and I decided unilaterally to cut the section on arc length from our Calculus II course: it was one lecture, we did not use it later in the course, students were just memorizing the formula, and it does not lend itself to concrete calculations outside of a few mostly artificial curves. So we cut it. All was fine… until the next semester when the Calculus III instructors wrote to us and asked if it was really true we had not taught arc length integrals. It turns out they were teaching arc length of space curves and told students it is similar to what they saw for arc length of plane curves in Calculus II. The students then said they had never seen arc length integrals in any form before. Oops! My department then created a policy that service courses could not cut topics from the syllabus without clearing it first with an undergraduate course committee.

$\endgroup$
2
  • $\begingroup$ +1 from me. I do question whether the omission of one lecture can do any real damage though. I think if a student has understood "the integration idea" then they should be able to derive the formula for arclength. This one is trickier than most, however, because of the need to utilize the derivative to approximate the "rise" along each section. $\endgroup$ Feb 10 at 22:40
  • 4
    $\begingroup$ @StevenGubkin most applications in a calculus course, in my experience, are covered in one lecture since they are taken from one section of the book. What does "real damage" by not covering a topic is that the students don't do any actual homework problems on the topic and thus don't have any experience working with it. When a later instructor wants to build on an earlier idea and that idea is not in the background knowledge of the students, then that later instructor has to use up their own class time on the missing lecture and their own planned lecture. $\endgroup$
    – KCd
    Feb 11 at 2:32
8
$\begingroup$

It is usually covered in Calculus II, I believe. I do a little of it at the end of Calculus I, because it is so beautiful, and it feels like a perfect closing to the show.

Useful? What in Calculus I is useful as it is? It's the concepts that are useful, and this is a new concept for the students.

Do they know why the volume of the cone uses a 1/3? Probably not. And it is such a simple problem, once you have the tool of volumes of rotation. Beautiful.

$\endgroup$
2
$\begingroup$

It's a forum style opinion, but I recommend to teach volumes, not your special topic. Sorry, no Euclidean proof, but will at least blabla about why I feel that way. Maybe it helps you get the heart to do it.

  1. As far as location (calc 1 or 2), I think 2 makes more sense as it is a little trickier topic. Also, the Naval Academy (very traditional) uses Stewart in order and has this topic in their second semester (their syllabus online, not sure how to link a PDF though). Really, I would coordinate with department unless you are teaching 2. If it's in a chapter of Stewart you are supposed to hit ("get through #X chapter"), than that sort of answers your question.

  2. As far as just not doing it ever...and substituting a topic of your own interest, I wouldn't. Teach the traditional course. This is an entry, service course. Things are hard enough for you, for the kids...might as well try driving down the middle of the road. Spend your efforts, ingenuity on pedagogy, training (they are non trivial)...more than on changing the content.

  3. FWIW, I liked the topic (granted I was a strong student and my classmates strong also).

  4. Would still prefer it to numerical topics. Many students are/were like me also. The special enriched sections of calculus with computers were reviled, for instance. Was better to place out...or take the normal track.

  5. I don't think it's a most crucial topic ever. In the sense of say max/min optimizations...or knowing that acceleration is the derivative of velocity. Like if this was calc for business or for nurses or the like, sure cut it...and tricky integrations also. But that's when you are generally watering down. Not when you are going off the reservation to do a numerical topic.

  6. I think there is some general benefit in getting students to think about more complex problems (just in the sense of things to track, manipulations). Also, I'm not even a visual person...but it was good for me to think a little about 3-d shapes. And for engineers and the like, good for them also. I mean Joseph's illustration is beautiful.

  7. Also sort of a "bridge to calc 3" type topic, maybe not directly...but in spirit. Moving out of the plane. Also physics 2 E&M involves some spheres and weird calc 3 style integrals. Not that I recall ever solving a volume. But still...kinda next door.

  8. I do appreciate that you are reacting to your own experience. And in the absence of anything, it is a Bayesian input. But I would not overly bias a sample of one. Also, you may get a different feel, love, comprehension of the topic after teaching it.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.