Why do we care so much about solids/surfaces of revolution?

Question

Problems involving solids/surfaces of revolution seem to be a fairly standard part of any calculus curriculum (at least in the USA), but the topic is so incredibly specific that I can't think of any motivation that doesn't involve pottery wheels or lathes. Of course pottery wheels are the basis for civilization, but I'm not sure that lathes are so essential that they should be considered an integral (hah, integral!) part of calculus, on par with, say, absolute convergence.

Either I am vastly underestimating the importance of lathes (which is likely, because I know nothing about manufacturing) or there is some other significance that solids/surfaces of revolution have to mathematics. To be honest, I don't think I've ever seen the subject brought up outside of a calculus class (aside from CAD and the whole lathe thing) - not in analysis, not in topology, not in algebra or number theory.

I just think it's weird that I have three calculus textbooks with more than one section devoted to the subject. Math SE even has tag for it! Is it really just pedagogical with no significance beyond "this is something you can use integrals for"?

Answer 1

One answer is certainly that these are objects that we can analyze using single-variable calculus techniques, and so textbooks and curriculums include them because they can be done.

But I want to add to this perspective: it's one thing to say "(single-variable) integrals can be used to evaluate volumes and surfaces" ... but which integrals, exactly? It's an important skill in calculus to be able to derive the precise integral required to calculate a particular quantity. Usually that derivation proceeds by approximating the quantity by finite sums (of lengths of line segments/areas of rectangles/volumes of cuboids), taking the limit as the number of approximants goes to infinity, and recognizing the resulting limit as a Riemann sums limit that defines a particular integral.

So teaching students how to calculate volumes and surface areas of solids of revolution is about more than just learning the formulas for calculating those quantites. It's about teaching students why such formulas are valid and how similar techniques can be used to derive new formulas. Solids of revolution are a way of teaching those more theoretical skills in single-variable calculus, so that students will see it earlier in their careers.

Answer 2

Solids of revolution would typically be used to calculate the volume or moment of inertia of a solid body, which would be most relevant in engineering design, which is quite a different thing from machining or from pottery. It's unlikely that a ceramic utensil would be spun at high enough speeds that one would need to know the moment of inertia, although apparently physicist Richard Feynman was inspired to come up with one of his theories after watching someone spinning plates on his fingertips.

It's in calculus books because it's a simple extension of two-dimensional ideas into three dimensions with an axis of symmetry. One can use 2-D calculus for 3-D problems, and also learn a useful method of computing volumes etc. of rotationally symmetric objects.

Answer 3

Surfaces of revolution are certainly very common in everyday life. As you noted, anything made on a lathe will consist mostly of surfaces of revolution. Also, any shape made by drilling or boring will be cylindrical, and countersinks are always conical — both surfaces of revolution. The cutter of a milling machine is often driven along a circular path, and this will again produce surfaces of revolution.

Solids of revolution are common in CAD software because these systems reflect the real world of engineering and manufacturing. Revolved shapes are useful in design because they’re easy to describe. They’re only 2.5D, so once you’ve specified the cross-sectional curve, you’ve specified the entire shape. This makes for easy description on traditional engineering drawings.

So, objects that include surfaces of revolution are very common. However, objects that are “pure” solids of revolution (consisting entirely of co-axial surfaces of revolution) are not so common. Even a part made on a lathe will typically have a few holes drilled in it, so it will no longer be a pure solid of revolution. And, of course, a coffee cup made on a potter’s wheel will usually have a handle. So, in practice, formulas for volume and areas of pure solids of revolution are not much use.

Does the prevalence in everyday life explain the attention given in calculus textbooks? Probably not, since people who write these books often have scant knowledge of engineering and manufacturing. I suspect that calculus texts like to talk about solids of revolution because these are more tangible and more interesting than graphs of univariate functions — they make calculus seem more real.

Answer 4

The amount of time given to solids of revolution is one of my many complaints with the AP Calculus curriculum. One would expect that a topic that so much time is put into would be highly relevant to engineering, mathematics and science.

The truth is that it is not. (At least for me, I never had to calculate the volume of a solid of revolution in my ENTIRE undergraduate applied mathematics degree.) Its only pedagogical significance, as far as I can see, is as you say - "something that you can use integrals for".

In fact, I think that the time spent on solids of revolution is actually destructive, not constructive, when it comes to teaching students about single variable integration. You take a student who has only just learned about integration in one dimension, and, with no regard as to why, thrust them into a topic that can only be well understood if the student already has somewhat of a foundation in higher dimensional calculus (polar coordinates, volume elements, etc). What does this do? It shifts the difficulty of the problem away from the actual integration and instead places it on visualization. A useful skill for sure, but, again, better reserved for when the student finds him or herself in a multivariable calculus course. Instead of getting into deeper topics in single variable calculus and real analysis, the student will spend countless hours diddling around with the "disk method", "washer method", and "shell method", none of which are useful skills in any other context than crunching out answers to stupid solids of revolution problems. All of these methods will be (and should be) promptly forgotten the second the student completes the AP exam.

It also only reinforces the naive notion that integration is only used to calculate areas and volumes. I have seen an innumerable amount of questions on integration on this website, usually in the context of multivariable calculus, that are somewhere along the lines of "Where is the area that this integral is calculating?" Or "How do I visualize this integral?" or "what does integration have to do with areas and volumes?". As anyone with a decent level of mathematical rigor knows, the integral is nothing more than adding up the values at all of the points in some set. The connection to areas and volumes is nothing more than our human way of prescribing some physical meaning to it. Unfortunately, the way mathematics is commonly taught, students have a hard time breaking the imaginary bond between integration and areas.

As a final point, this is unfortunately just another example of a theme that is common in grade school and lower college level mathematics. It is easier to write, and grade, bad problems than it is good ones. It is far easier for an AP grader to notice that the student forgot a factor of $\pi$, or forgot an $r^2$, or used the washer method instead of the shell method, or any other trivial mistake, than to go through an attempted proof of the mean value theorem to find a logical error.

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At best, the solids of revolution are a misguided attempt to make math feel more "real". The students who hate mathematics will not care what the context of the problem is. They really couldn't care less about any math problem, even if it's something remotely interesting like calculating the volume of a vase. And for the students that actually like mathematics, I think they would prefer to learn more new, interesting math, than to just dress up single variable integration in a fancy circular outfit and repeat the same boring calculations they have been doing for the last few weeks.

At worst, the solids of revolution is just a way to add a needless amount of complication and details that will induce more mistakes from students and hence make it easier to put them on a bell curve for scoring purposes. Given the reputation of the College Board, I am far more inclined to believe the latter than the former.