Why do we need to learn integration techniques?

Question

After a lifetime of approaching math the wrong way, I took two college math courses this quarter with a newfound zest for math. These classes are integral calc and multivariable calc.

Integral calc started out okay, learning about Riemann sums and the Fundamental Theorem of calculus. But instead of spending a great deal of time gaining the intuition behind these things, we jumped into integration technique after integration technique.

Why on EARTH would we need to memorize and regurgitate a bunch of integration methods on toy problems for 6 weeks? It's absolutely bizarre. I'm not taking this at some random JC either, this is at a top research university. This class is single-handedly destroying my enthusiasm for calculus.

When we have software that can do much more difficult integrals than we can with pencil and paper, why would we waste time memorizing Trig substitutions or integration by partial sums? Doesn't that just make it a glorified algebra class?

Answer 1

This is, in my opinion, a common feeling after a "lifetime of approaching math the wrong way". People are taught math in a very rigid rule-based formula/pattern method, and then when they contrast this against mathematical proofs they have a knee-jerk reaction against anything which looks even remotely like what they did before. The fact of the matter is, however, that you will need to be able to do some of this without aid of a computer.

When reading a proof, it is easy to take for granted that you are able to fill in the details between steps of the proof, when really all these steps are able to be filled in precisely because you have the understanding of solving equations (basic algebra) and working with inequalities (basic arithmetic), for example. The same thing holds true for proofs involving integration and derivatives.

Perhaps it's best to leave it to those who really know what they're talking about - Spivak writes in his chapter on integration that our motivation should be that:

1. Integration is a standard topic in calculus, and everyone should know about it.
2. Every once in a while you might actually need to evaluate an integral, under conditions which do not allow you to consult any of the standard integral tables.
3. The most useful "methods" of integration are actually very important theorems (that apply to all functions, not just elementary ones).

He emphasizes that the last reason is the most crucial.

I would personally advocate that students should be wary of falling into the trap of thinking that such pedantic methods are beneath them. It is often easy to think you understand something at a high level, but you don't truly learn what it is all about until you really get your hands dirty with it.

Answer 2

A short answer from Isaac Asimov : 'The Feeling of Power'.

Answer 3

Here is a little fable. Some parts may resemble actual events that occurred someplace or other. Other parts are purely made up.

A large university offers different types of calculus courses for the benefit of various other departments. One day the Mechanical Engineering curriculum committee comes to the Mathematics curriculum committee.

"Look!", they say. "If we tell prospective students that we require $X$ semester hours of math for our major, but a competing university says they require only $X-2$ hours, that other place will win the students. So we have to cut $2$ hours from the required math courses."

"OK," Math says. "What do you want to cut?"

Mechanical Engineering studies the syllabi, and says (among other things), "Cut out techniques of integration. Students can do all that by computer, anyway."

So the new--sleeker--course for Mechanical Engineers comes to be.

A few years pass.

Big important Senior Professor is teaching a course for Mechanical Engineers. He derives an equation for this problem he is doing. Then he says, "Now we integrate by parts to put the problem in this other form, so we see it means that we should minimize the energy."

The students reply: "Integration by parts? We've never heard of that!" (Actually, integration by parts had been mentioned a couple of times in their math course, but they had no baby problems to practice it on, and thus they have forgotten it.)

NOW Mechanical Engineering is accusing Mathematics of shoddy work... The members of both curriculum committees have changed by then, so neither department is likely to remember the reason that techniques of integration was omitted from the integral calculus course.

Answer 4

Somebody has to play devil's advocate here, so I guess it's up to me.

The second semester of freshman calc is universally devoted to force-feeding students with tricks for integration which they will neither need nor retain after their final exam. This wouldn't be so bad if there were some other educational purpose being served. For comparison, a student taking an English class might neither need nor retain any specific skills or knowledge related to the novel White Teeth by Zadie Smith, but the student is arguably developing skills such as critical thinking and written expression. There is unfortunately no such argument to be made for techniques of integration. Students learn this bag of tricks as a set of cookbook procedures -- the opposite of critical thinking.

The analog of the skill in written expression taught in an Engligh course would be facility in performing integration. This analogy fails, however, because written expression is a skill that can only be performed by the human mind, whereas computers can now carry out integration so well that the human who can outperform Wolfram Alpha in integration is as rare as the human who can (on a good day) win a game of chess against Deep Blue. Since computers don't seem to have attained creativity or self-awareness, it should be deeply demoralizing to the practitioners of any art when their craft is computerized. It shows that what they were doing could have been done better by a mindless automaton. For this reason, it's unlikely that humans a hundred years from now will be much interested in chess or integration techniques.

In the nineteenth century, men who wanted to become officers in the British military were required to demonstrate proficiency in ancient Greek. The intention was to exclude the working class. Today, people who want to be doctors are required to demonstrate proficiency in techniques of integration. The intention is to weed out students from the "impacted major" of biology at the University of California. In defense of the nineteenth-century British ruling class, we can observe that translating English into ancient Greek is still a skill that can't be carried out by a computer.

Answer 5

It is good to know how to integrate functions even for those three reasons:

1. You do not have to turn on your computer to check every triviality.

2. You can check if a black-box program is not giving you total bollocks.

3. You can amaze people by actually showing how to get formulas for cones' volumes, bottle volumes etc...

Answer 6

I would argue that the purpose of these techniques (to a mathematician) is not so much the value of integrals but how the integration techniques enrich advanced mathematics.

Typically the techniques studied are integration by parts, trig substitution, and partial fractions. It would be possible to enumerate countless uses of these ideas in the literature, but I will restrict myself to only a few.

Integration by parts shows up all over the place in mathematics, especially in differential equations, which will be an important course for anybody studying math. The first place you'll probably run into it there is doing Laplace transforms, and especially when proving properties of the very important gamma function (for example, you can prove $\left(-\frac{1}{2} \right)$ factorial is $\sqrt{\pi}$).

If you happen to go on farther into partial differential equation theory, you'll study Sobolev spaces which are roughly spaces of functions who have partial derivatives, but they are actually weak derivatives which are defined in terms of integration by parts. Sobolev theory is exceptionally abstract and it would be a huge pain to have to introduce integration by parts before studying them.

Partial fraction decomposition will also be used in differential equations to find inverse Laplace transforms. It's not the only way of doing this, but you'll see them there for sure. Again, it would not be advisable to introduce partial fractions here, since inverse Laplace transforms are already difficult enough deep into a DE's course.

Trigonometric substitution forms a good base for studying elliptic integrals, especially Jacobi elliptic functions.

So really, the point isn't to make it so you can compute $\displaystyle\int_0^1 xe^x dx$ (though you should be able to or you look silly!), but rather so you are able to comprehend more advanced and abstract things that use these techniques as part of their definition or part of their theory in some way.

Answer 7

Perhaps you might like this TED talk.

http://www.youtube.com/watch?v=60OVlfAUPJg

It is a very interesting question and I am not exactly sure where to draw the line, but it has become very clear that learning to use a high-end CAS will have amazing benefits in the learning experience, IMO.

Answer 8

Why do we need to learn integration techniques?

Because it is very useful and is the foundation for higher mathematics.

Personally, I never understood the objective of maths lessons until I started solving real world problem in my working life. I realised that one important lesson to be brought home from these "boring" lessons is not how to do integration or calculus itself, but rather, to be able to spot integration and calculus problems in real life and then formulate a mathematical statement about the problem and then use a computer to give you results.

I am currently a software engineer. I use calculus for programming graphics and animations; cost optimisation for manufacturing; and provide quantitative financial analysis on graphs and charts. There may be various methods and conceptual approach to these problems, but realising calculus is also one of them can help a lot.

Answer 9

Applying the same reasoning, why do you talk if a computer also can do it? In one word: independence. Computers run algorithms created by humans, but, what if one day the computer doesn't know an answer to your integral? What if it's badly programmed? What if the computer breaks? Mathematical problems in life would still there. Who would solve future problems? What would you do?

Answer 10

I believe knowing different techniques does provide you with familiarity with commonly encountered expressions and techniques in other situations. For example, the partial fraction decomposition is important whenever you see a linear transformation. Integration is just one of many linear transformations. Green's identity (integration by parts) is also super-important, and there are several occasions your math software cannot help you with this. The less important topic could be the half-angle tangent substitution, but it's applicable in so many cases, so why don't we just memorize this one technique, right?

Anyway, I agree that 8 weeks of just integration may not be a good thing. If you don't see how the techniques are used, you may not realize their importance, and forget them eventually. That happens a lot. Maybe the course is organized like this for the convenience of the teachers and TAs(?) Just my 2 cents.

Answer 11

It seems to me the initial qualm is with the lack of understanding provided with practice. If you do not understand why 2 + 2 = 4, why practice it? Just use a calculator. Practice only becomes effective when you know the reasoning behind what you are doing and why the method works. A lot of times, in math courses, you are given tips and tricks to practice without ever being taught how they work or how to look for situations where they become useful. It is correct to say that you do not need to memorize things once you have a true understanding. Why introduce topics to memorize and not provide a true understanding? This seems to be the OP's issue.

Answer 12

The problem is that once you hit Ordinary Differential Equations a lot of those turn out to be needed, in fact the specific ones that are included is almost certainly based on "ones we will need later". In a very difficult course for students like ODEs if you take time to introduce a new integration method (because you didn't already cover it) you will make the course impossible for the vast bulk of students because there would be too much to take in.

Some integration techniques will come up in one form or another again and again (substitution and integration by parts) in other courses and need to be as natural as multiplication and division for advanced mathematics. The more specialized ones you mention (trig substitution and partial fractions) also play the role of forcing you to do enough practice with using the fundamental ones as part of a more complicated problem to ensure you really get serious practice with them, i.e. can use them in a context bigger than a single problem where that is the only thing you have to do.

Answer 13

I think there are some students like me who want to study math in way that each statement is examined like how it got here what axioms/hypthesis caused it to be true, what if an axioms is removed then how it will effect on that statement. Instead of giving ready-made theorems, I would rather try to prove in timely fashion with some intermediate theorems as hints. Many of us want to (re-)discover it, rather than learn it.

So when these integration techniques are presented and the students hold-off, it just takes the fun away, not because you can't do it in your free time but because the free time is way shorter than the time needed to go thoroughly into such vast subject. so the choices you are left with is either curb your enthusiasm and pass the exams with the question "Do I really understand it?" or fail the exams with the smile on your face and satisfaction saying "That felt good".

The point is to devide a subject more flexibly so that no one feels force-feeded. So what if someone doesn't feel that something is not important, force feeding won't benefit him or the society. Instead letting people do what they want to do would likely help the advancement in their life and the society itself.

There would be people who want use math solving program and there would be ones who want to create it. The point is (again), that people from both side will exist. Even if one is in majority. so what @ordinary asked does he have to learn the intergration methods even if the same can be done by a software or the methods is not going to be needed in future? Even he belongs to the minority (the math-rediscoverer), he does not. Maybe he will realize otherwise in future or may be he won't? If he does, there is nothing wrong with going back and read/learn it but this time with motivation. If he does not, time well saved. So I think it is best to leave it to the future. No point in force feeding now and even if people like @ordinary become majority (may be they are), it does not mean the other side will go extinct?