For some reason, in calculus classes here in the U.S. we spend a lot of time teaching students how to find the volume of solids of revolution. This is invariably confusing, because often students try to memorize the various "methods" (disk, washer, cylindrical shell) and have trouble getting a handle on the actual geometry. When I've taught this, I've encouraged my students to draw lots of pictures. This is more than I can say for some textbooks, which try to give "rules" of the form "if a curve is entirely above the x-axis, and it's being rotated around the x-axis, between y = a and y = b, then here's the formula" -- I don't recall explicitly knowing that students have tried to memorize such a formula and failed, but I wouldn't be surprised if a lot of the wrong answers I've gotten from students on questions like this are based on such things.
Now, as a probabilist, I do have some use for calculus in my work. But I can't remember the last time I needed to know the volume of a solid of revolution.
Then again, my work is not particularly geometrical in nature.
So, I ask -- why do we spend so much time on this? Is this something that students actually need to know? I'm inclined to guess that it's just tradition. But at the same time I can't rule out that solids of revolution are actually so prevalent in engineering and physics (the traditional "customers" for the calculus course). Also, a lot of the "standard" calculus course seems to be a sequence of contrived problems that exist basically to make the student do various derivatives and integrals.
A random walk through mathematics -- mostly through the random part.
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33 comments:
I wonder if it has to do with the fact that historically, these sorts of calculations were one of the triumphs of calculus. Before calculus, it was possible to calculate volumes of some solids, but there was no systematic approach. Perhaps the emphasis is inherited from the "hey look what I can do" excitement from a few hundred years ago.
Jesse,
perhaps that's true. I do remember being excited that I could calculate these things when I first learned how.
I think one reason is that it is possible to visualize solids of revolution, so you get a clear conception of what's happening. In the future, if you want to reason out how to solve a problem, you can map the problem onto the geometric problem (which you have an intuition for) and solve that.
The undergraduate course “complex analysis” in most places is not much more than a litany of integration tricks, the semester-long version of learning solids of revolution.
The professor I took the course from got to the end of a particularly involved proof that some trick was correct for a particular special case and turned reflective, then sad. He lamented that no one really needed to learn this anymore, because now we have Mathematica. Another student in the class attempted to cheer him up, pointing out that someone has to program Mathematica.
This worked for a couple of seconds, but then he replied, “No. Mathematica has already been programmed.”
Now, as a probabilist, I do have some use for calculus in my work. But I can't remember the last time I needed to know the volume of a solid of revolution.
Prove that the normalization constant for the normal distribution is $(2\pi)^{-1/2}$.
I remember the realization that you can fill the bucket with paint, can't paint the outside of said [infinitely thin walled] bucket].
The whole solid of revolution thing is kind of silly because it gets away from the central idea. An integral is the sum of a bunch of small densities. Grokking all the ways that a density could be represented is, practically, more important than learning to subtract and multiply for the typical calculus student.
I assume you mean this in the context of first-year calculus, not multivariable or vector calculus. I think part of the reason for discussing surfaces of revolution is that they are one of the few kinds of solids whose volumes can be computed without introducing the the full machinery of multivariable calculus. (Even though it is, of course, used implicitly.) It's also an interesting application of integration, without the need of a lot of background from e.g. physics. In addition, it gives the students a little practice at thinking 3-dimensionally, which lessens the shock when they take multivariable calculus. After all, dealing with surfaces of revolution pales in comparison to setting up a triple-integral over some more general region.
I am an AP Calculus teacher, so this is one of the big things on our list to do, too.
Basically, it is useful for getting back to the true nature of integration. That really you are summing over smaller parts. It's helping students see that this can be done with more than just "rectangles" and "trapezoids," but in higher dimensions, too. So, it is as an introduction to the multivariate in parts.
At least in the AP class, we also do other types of solids with similar cross-sections (pyramids, something with a parabolic base and equilateral triangle cross sections perpendicular to the base, etc.). So, students are supposed to see that integration is really just an infinite sum and not just anti-derivative.
It also adds a little "application" to a somewhat abstract idea for high school seniors.
As for the book methods of memorizing formulas and such, I've found that a majority of the students really want to use those instead. The concept of finding cross-sections is difficult, but knowing that cylindrical shells will always be 2(pi)r h is comforting to them, I guess. So, it ends up losing its meaning and becoming another exercise.
unapologetic,
I usually see that by squaring the necessary integral and changing to polar coordinates. How can I use a solid of revolution?
Boris: you're considering the integral of exp(-(x^2+y^2)) over the xy-plane. The usual way to do this is to change to polar coordinates. But the volume between the graph of exp(-(x^2+y^2)) and the xy-plane is in fact a solid of revolution, namely the solid obtained by revolving the area between the x-axis and the graph of exp(-x^2) around the y-axis.
Calculus Dave: you make a good point about doing this with various other cross-sections (and in the course I've taught, this comes up). But the same complaint occurs to me -- when are you going to have to compute the volumes of these things? I suspect that the value here is really in getting students to think geometrically and to conceptualize the integral as an infinite sum.
I agree with "D" that the point of covering volumes of revolution is the derivation, not the result. It would help if textbooks and instructors would emphasize that.
As for the paint bucket that can't hold enough paint to paint itself, this isn't as unrealistic as it sounds. You could actually create such a container. Because of surface tension, the end of the container would have no paint inside. Or if you want to consider an idealized liquid with no surface tension, you could point out that painting the outside implies a certain minimum thickness.
Isabel is right, Boris. "Converting to polar coordinates" is the exact same thing. The whole point of solids of revolution is that they look really nice in cylindrical coordinates, so we can do two of the three integrals necessary to find a volume without using antiderivatives. Then we leave the last one as a single integral for the student to do as the work in the problem.
And I make a big deal of this when I teach calculus 3: these formulæ from calculus 2 are all just special cases of the techniques we just learned.
It turns out that the railroads have a requirement that loads attached to flatbed railcars be loaded with their center of mass centered on the car. This is to prevent trains from fallin over, I guess. So I used calculus once to figure out the center of mass of a half cylindrical shell. That's the only use I can remember from my blue-collar career.
The whole concept of all these formulae in math is completely ridiculous, I think (though I am abnormal and very verbal). I realize now this is why I hated math in loved physics in high school. In physics the formula naturally flowed from the physical properties we sought to describe. That's why when I first saw calculus, I just repeated over and over in my head "acceleration is the derivative of velocity". Now that I'm trying to go back and learn relearn all the math I never bothered to learn before, I can see it’s all just a bunch of adhoc special case formulae to memorize. You can sit through an entire semester of Trig and still not know what Sine is. Is just a black bock, you know when you put things into it what pops out, but as soon as you forget all the formula you memorized you're back to not knowing anything! (I would say you didn’t actually know anything to begin with, in retrospect) How many years do people go on knowing pi is~ 3.14 and having no idea it is the ratio between the circumference and the diameter? Ask a student who’s taken one semester of (school mandated) statistics, what a standard deviation is, I’m 95% confident if they give you an answer at all it will be a formula and if you ask the student “yeah but what is it designed to measure” they’ll just stare at you blankly.
The whole freakin system is out of order! I’d like a math book that read like a history book, starts with the most basic problems people we’re able to understand, tells the story of how they gained that knowledge and what properties they discovered. Then moves on the next advance from there, at least, I would have enjoyed this, I wish it were at least an option….though I guess it would take too long?
P.S. As for the normal distribution, it feels like and exponentiation of Cosine to me, I try to figure out how de moivre came up with it when I get bored sitting in my statistics class, so far I’ve got y=exp(-2*pi*x^2) i'm not sure how the root-mean-square got into the mix.....though
kurt: what are you talking about? Every trigonometry curriculum I've ever seen starts by defining the sine and cosine in terms of triangles (and the unit circle). That's why it's trigonometry.
Now, if you didn't pay attention... but we can hardly fault the course for that, now, can we?
right exactly, your memorize the ratios of height over hypotenuse for a few easy to remember angles and you can know if you see Pi/2 the sine is 1 but you don't actually know what Sine is doing in the "black box" if someone gives you an angle you didn't happen to memorize then you pile on more ad hoc formulae for Sine (Something I know + something I don't know) great of I can solve the problem. But how about they just give me 1 formula for every possible input to sine and I’ll just use that. In instead of memorizing half angles, double angles, power reductions. Why is this necessary? Sure it’s built into my calculator, but its still a black box. I'd like to see an actual equation with just F(x) and X's.
I see the Taylor series for Trig functions over there on Wikipedia hanging out, why can't they just use that?
Kurt, why do I get the idea you're trolling here? You say that students don't understand what the sine function is even when it's defined geometrically (meaning we can get our hands on it and reason about it without focusing on the calculations you're so obsessed with), and you propose replacing it with infinite series?
Isabel and unapologetic, thanks. I missed that. =)
I'm not trying to "troll". I don't know what "students" understand. When I was in high school, i didn't bother to learn any math because I didn't think it was useful and I just did enough to pass, when I was in college I got A's in math but taking non-major courses.
I am now going back and trying to learn some math because I've found it has some useful properties. I find reading about the properties and the reason they originally were discovered, by whom, when, what discoveries proceeded and what discoveries followed is quite helpful to my understanding of the nature of the concept being explored. I'd rather read a 20 page essay on the function and do 10 problems using it than read 3 pages and do 30 problems. I feel like i learn so much more from reading 1 research paper than an entire chapter of a math book.
I want to understand what is happening not memorize 16 rules which are connected in someway which the instructor does not see fit to share with me, those 16 rules are already in my calculator, if when I get done and I still don't what the properties of the function are. what is that point, to save on batteries?
But as I said I am an odd bird, and I am verbal. Maybe it is precisely to avoid such explication that others study math.
Kurt, the reason no one gives you the 'rule' is that the rule is completely unusable. The Wikipedia page gives a number of different ways of defining the sine function; most of them are the ones we've brought up. I personally like the unit circle definition. But if you want a formula, you can find it here. The problem is that it's an infinite series and almost impossible to work with effectively, and it will never give you an exact answer. But if you want it, there it is.
It's because "Centroids of Solid Revolution" would be a great name for a rock band.
Long ago I used to have a match teacher eplaining that trigonomertry and such were very useful to us Dutch, while roaming the seas, finding out where we were shooting stars. Generally speaking I feel the only use of any math is that it shapes the mind into logical/abstract thinking.
So Kurt, just to be clear: you had already decided not to pay attention to math in school because you thought it was useless. As a result you didn't understand the definition of the trigonometric functions when they were presented, and you only remember the formulæ which followed (which are examples of reasoning about the functions from their geometric definitions rather than in terms of calculational rules). And now you want to blame the course instead of your own disinterest.
I did a lot of this in several calc classes (with the same prof for 1 thru 3). To try answering Isabel's question in the post, my view was always that these calc classes are often the branching point when some people decide to head off towards engineering, and some head off towards something like pure math. Aside from DiffEq, the engineers I know didn't really have the same math curriculum as the math students, save for these early classes in calc. So the calc class seems to have to serve those who need it to help with intuition on "physical" things, and as a starting point to understanding analysis for those who will study math as math. Sure, it's not a clear line by any means, but I think it just may be that early calc has always had to serve multiple kinds of interests, so things like "solids of revolution" are kept because it's a decent example that may either be relevant later on, or not.
Kurt,
Assuming your are being serious, there is a book you might want to get. It's an old one that I have that is probably out of print. But I'm certain you can still find copies available on the internet. It's called "Mathematics for the Millions" by Lancelot Hogben. I really enjoyed reading it.
Strangely enough I have seen mention of this book several times during the past year while looking at a wide variety of sites. David Mumford from Brown University mentions it in an article you can find on his website listed under 'articles of general interest'.
Hypatia
I am now going back and trying to learn some math because I've found it has some useful properties. I find reading about the properties and the reason they originally were discovered, by whom, when, what discoveries proceeded and what discoveries followed is quite helpful to my understanding of the nature of the concept being explored.
I think Kurt is emphasizing the fact that a lot of times examining the history of a concept, say, that of a function, can go a long way in motivating one to study that concept with much greater "zeal". All of mathematics today is axiomatized, which is good for the subject, but from a pedagogical standpoint it tends not to elicit interest in the minds of beginners/novices. In high school, as well as in college, almost no emphasis is placed on the progression of ideas and how they interact with each other, leading to the discovery/creation of new concepts in mathematics. Of course, one could argue that it is the student that must work hard to keep his/her level of motivation high, but I also think that teachers can do a much better job in presenting the history of mathematical ideas in a way that can genuinely arouse interest in the minds of students about the subject.
Also, I think Kurt is saying that had mathematics been presented in high school the way he wanted them to be, then he would have had learned much more, and I strongly believe that there are a lot of students who think the same. In the end, it really boils down to good pedagogy.
Unapologetic, I'm saying that I decided not to pay attention in high school math because I saw no use for the math and intended to avoid any career in which math would be necessary. I quite easily remembered the definitions of the trig functions because they were useful for solving physics problems, which I enjoyed. However, the math itself was of no interest to me for its own sake.
Now I have taken the course again! And looking back as a person who is interested in the subject now, and very committed to learning, I can see exactly why I hated it in high school. As topological musing points out, its the axioms. When I was in high school I wouldn't have bothered, to think about all of this, I would have just tossed it into the "this is stupid" category and used my calculator. Its only now that I'm studying the subject intently that I can see that all the axioms are a serious problem.
When you question my "disinterest" as a young age you assume that I would automatically be interested in something just because, it is presented to me in school. As someone who later went on to study a bit of economics I look at things from a completely different perspective. My time is valuable, there are a limited number of things that can be read and studied, each (required) class that I take I approach from the perspective that it is a waste of my valuable time, until the instructor has demonstrated to me that their subject is worth my attention. History, English, physics, they are all constantly justifying the validity of their usefulness to you in the class. Even when when their validity is dubious (most of the humanities). I am not blaming the subject for my ignorance, I openly say i chose not to learn. But math must compete in the market place of ideas with other (lesser) subjects. And yet no one ever made a case to me for why the subject matter was important. This would simply be my problem if America were producing plenty of math students, but i feel that I'm in the majority here, people are proud of saying they're "bad at math" in this country. And, what I am trying to say is I think an alternate option in the the approach to teaching the subject would go a long way.
Somewhere, on this blog I think, there was a statement that many students become graduate students in math only to find out that they're not good at math, they're good at following instructions. I am one of those people who never follows instructions to assemble things, the type of person who just dumps everything on the floor tries to infer how it must logically fit together. I suppose that's the problem here. But, again, from an economic perspective, we are customers too! Someone somewhere should be presenting the material in way we find accessible!
P.S.
The unit circle is a beautiful eloquent logical extension.
There's nothing logical about memorizing sheets trig identities.
I'm not going to say that any teacher is great, and you may well have had a bad one, but you're indicting the entire trigonometry curriculum because you had a bad time of it.
So we take out the axioms and put in a history lesson. What, now, happens to the students who would have learned better from formal rules? And what happens in either case to students who'd do better cutting and pasting physical scraps of paper?
Teachers and curriculum designers have limited resources. I'm not an expert in secondary mathematics education, and I'm not willing to second-guess the trade-offs made by those who are based on anecdotal evidence.
"Teachers and curriculum designers have limited resources. I'm not an expert in secondary mathematics education, and I'm not willing to second-guess the trade-offs made by those who are based on anecdotal evidence."
That's how I feel about the axioms. I'm not going to believe these things are useful or even true unless the instructor demonstrates why and how this is unnecessarily the case. So that I if forget the formula I can derive it from knowledge I already have.
P.S. this "Mathematics for the Million" seems like a very good starting place, thanks Hypatia
unnecessarily=necessarily
I think solids of revolution are just nice application problems, nothing more. The washers, disks, and what's the third "method"? unhelpful.
Here we have something to visualize, not too tough, how are you going to slice your solid so that the dimensions of the slice are easy to calculate and add up? Which is going to be your "thin" direction? How far do you need to go (limits)? and can you take advantage of symmetry?
Jonathan
Some more references for Kurt:
Calculus: A Genetic Approach by Otto Toeplitz.
Basic Calculus: From Archimedes to Newton to its Role in Science by Alexander J. Hahn.
Analysis by Its History (Undergraduate Texts in Mathematics. Readings in Mathematics) by Gerhard Wanner and Ernst Hairer.
Free Calculus: A liberation from Concepts and Proofs, by Qun Lin.
The Feynman Lectures on Physics, by Richard Feynman.
I teach non-AP calculus to students, and I agree with a lot of what has been said... I agree with you though, I don't recall ever seeing it or using it in college.
But there are two things which I think it's easy to overlook.
First is what Calculus Dave said in the comments. The pedagogical reason.
Second is the fact that at least my students seemed to be more interested/excited about it than a lot of the other calculus topics -- they could actually "see" it work. And any math my kids are excited about naturally is something I want to continue. (My students aren't "mathy" students -- most are going off to college to be poets and the like.)
And even I remember thinking that it was insanely cool, when I took calculus. So yeah, maybe it's not the most used thing, but I think it has a purpose.
If we're talking about things that students will never see again, ever, let's do away with two column proofs in geometry. If we want to teach them proofs, I'm sure there is an infinitely better way out there that doesn't make you want to jab a compass in the back of your hand.
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