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This is a soft question.

  1. What is the purpose of teaching techniques of integration at the college level?

  2. More specifically, in the sense of putting integration into practice, what value does teaching these techniques serve in a world which increasingly depends on approximation and computer algebra systems?

  3. If the use of these techniques in practice is limited, then what should integral calculus be teaching in order to make the content imparted by the class more actionable?

I am assuming that, in general, the students of integral calculus will not go on to be mathematicians.

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  • $\begingroup$ Vote to combine this with all the other "use a calculator" and "use Wolfram Alpha" questions. And the retorts remain the same (see them for details). $\endgroup$ – guest Jul 25 '18 at 22:55
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    $\begingroup$ @guest, I think you are seriously misunderstanding my question. There may be a purpose to these techniques of integration. I don't see a clear answer to that question. One might retort with "polynomials are dense in $C^n([a,b]),$ then we can approximate the solutions . . . etc." What I would like are answers which provide a substantive, tangible, reason as to why we still teach these techniques. $\endgroup$ – Chickenmancer Jul 26 '18 at 0:29
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    $\begingroup$ Why do we teach literature? Why study history or philosophy? With Google translator etc. why study language? Maybe a better question to ask is why pay for your job training? To understand the derivation of physical theory and engineering calculus is very much needed. Try doing statistical thermodynamics without a good theoretical calculus background. Unless you're content being fed answers without derivations I don't see how it's going to work. $\endgroup$ – James S. Cook Jul 26 '18 at 2:05
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    $\begingroup$ Duplicate of math.stackexchange.com/q/188933/442 , from the time before matheducators.se existed. $\endgroup$ – Gerald Edgar Jul 26 '18 at 15:54
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    $\begingroup$ Since all integration techniques come down to substitution or by parts, I view them as a chance for students to really master basic algebra. $\endgroup$ – Aeryk Jul 30 '18 at 18:28
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There are many techniques of integration. Some of them, like integration by parts, are important theoretically. Integration by parts shows up in the derivation of the Euler-Lagrange equations in classical mechanics and in other situations where we study functionals on a domain with boundary.

Other techniques for integrals are there to complete the table of integrals of basic functions. It sounds like box-checking, but stats and physics can come up with some pretty wacky functions. That being said, I wish I had a non-artificial example of when $\int \sin^m x \cos^n x\ dx$ would come up.

Now, I think that it would be nice to have Applied Calculus classes actually be about using computers for these more approximate algorithms (e.g. Monte Carlo integration), rather than being slightly easier classes with more examples. (Hughes-Hallett does appox. integration by Riemann sums, but it is done by hand and with pitifully small datasets. Plus, no estimates on error are made.) However, I don't think that there is a market for that right now.

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    $\begingroup$ If we were only training mathematicians, it would appear that teaching students about error control, or having them appreciate that the error in a Riemann sum goes to zero would be useful. Error plays such a key role in analysis, and in real world engineering that it seems like a lost opportunity for both groups. $\endgroup$ – Chickenmancer Jul 27 '18 at 15:30
  • $\begingroup$ Actually there is an error estimate in Hughes-Hallett for Riemann sums of monotone functions. $\endgroup$ – Michael Bächtold Jul 31 '18 at 19:02
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    $\begingroup$ @MichaelBächtold I was thinking more of error estimates that you could use on realistic data. Rarely do we know a function is monotone. Ex: anything from econ. Also, more can be made of basic questions like "How many subdivisions do I need for accuracy to $\pm 0.01$?" $\endgroup$ – Adam Aug 1 '18 at 0:54
  • $\begingroup$ @Adam: What type or error estimates on realistic data are you thinking about? $\endgroup$ – Michael Bächtold Aug 1 '18 at 14:00
  • $\begingroup$ @MichaelBächtold For example, the error bounds in this: en.wikipedia.org/wiki/Trapezoidal_rule#Error_analysis or techniques for when the values are known to have noise. $\endgroup$ – Adam Aug 2 '18 at 1:44
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1: Up until the collegiate level, most math students memorize the 'right' way to do certain types of problems, and then repeat that hundreds of times on homework, so that they can repeat it on a test.

However, with integration there are too many possible forms to recognize and memorize each one. Instead the student is forced to try to figure out their own way of solving the problem from a common set of rules. Even though this process can often be automated, there are similar reasoning tasks, such as proof-writing, which are impractical to automate at this time. There are reasoning tasks in other fields such as programming and writing which also require the same kind of thinking: manipulation by predefined rules.

This does not always have its intended effect, as many students are stuck in the memorization-mindset, and it is easier for professors to say what to do without explaining why or what it means. There is (unfortunately) still much memorization involved, but it asks for more original thinking than prior math questions have.

2: For approximations, people try not to approximate things that could be computed directly to get better accuracy; approximations are usually for things that can't be integrated analytically.

Knowing what an integral really means (as opposed to knowing how to operate a CAS) is both useful and important even if students do not become mathematicians. The concepts of integrals and other infinitesimals lay the foundation for differential equations and stats. These are heavily utilized by all STEM fields, business, and the social sciences.

Rather, they should be viewed as two complementary skills that are valuable in different ways: learning how to use a CAS, and learning the theory (what it really means, why it is that particular way).

3: Maybe "integrate this function" problems gets boring and distant. It feels really useless if you could just type it into a CAS. Good professors will ask questions whose difficulty does not come from bookkeeping symbols but from figuring out what to do next.

Example: explain why the area of a circle is $\pi r^2$.

Try to answer this question on your own.

If you imagine the circle like an onion slice with concentric rings of width $dr$ (just a weird looking name for a small number), then you could 'unroll' each ring into a rectangle whose length would be the circumference of that ring $2 \pi r$ and width would be $dr$. Adding up all of the rings, and voilà $\int 2 \pi r\ dr = \pi r^2$.

Or maybe don't teach integral calculus at all

I have answered thus far as a 'Devil's advocate', and I am sympathetic to your viewpoint. I think there is some use to learning higher-level math so that students become more logical thinkers, but maybe not integral calculus. Integration involves more memorization than I would like, and ends up being tedious rather than rewarding.

I would be happy to see a number theory course be offered in place of calculus. Number theory asks questions like "how many prime numbers are there?” These questions sometimes have counterintuitive and interesting answers.

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  • $\begingroup$ I frequently (e.g., today) think about your exact example question, and how much more explanatory it seems when using the $\tau$ notation (direct visual connection with the integral of $r\ dr$; one less missing step). Related: Note the error in the original version of this answer that the ring circumference should be $2\pi r = \tau r$. $\endgroup$ – Daniel R. Collins Jul 26 '18 at 0:45
  • $\begingroup$ Indeed! I have edited accordingly. It seems that geometry prefers $\tau$, but analysis prefers $\pi$ (Basel Problem, Gaussian Integral, ) $\endgroup$ – charmoniumQ Jul 26 '18 at 0:57
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    $\begingroup$ I don't know of any reasoning tasks that provably "can never be automated" for which it's known that humans can do. $\endgroup$ – user797 Jul 27 '18 at 15:59
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    $\begingroup$ @charmoniumQ: It's trivial to write a computer program capable of writing a proof of any theorem -- simply iterate over all possible text documents, checking if the contents are a proof of the statement in question, and if you find one, output it. That we don't do this in practice is a matter of efficiency, not a matter of theoretical capability. $\endgroup$ – user797 Jul 30 '18 at 10:33
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    $\begingroup$ It deserves to be much better known that Turing designed his machines not as models of any type of physically realizable computer but rather as an ideal limit to what is computable by a human calculating in a step-by-step mechanical manner (without any use of intuition). See here for further discussion, including Gandy's generalization to machines with limitations based on the laws of physics. $\endgroup$ – Bill Dubuque Jul 30 '18 at 16:46
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Reposting a comment as an answer:

Since all integration techniques come down to substitution or by parts, I view them as a chance for students to really master basic algebra.

Corollaries to this line of thought: Don’t spend gobs of time on the techniques. (I take about 2 - 3 weeks total between substitution in first semester Calc and the rest in second semester.) And in later sections (surface area, volume, work, hydrostatic pressure, etc.) I focus on setting up the integral and assume/encourage them to use technology to actually get an answer when needed.

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    $\begingroup$ Thank you for reposting this. I can see the value in teaching them from the viewpoint of "teaching substitutions which reframe one problem in the context of another" like trig subs being helpful, too. $\endgroup$ – Chickenmancer Jul 31 '18 at 0:21

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