What's the deal with integration?

Question

So at uni we learned tricks and techniques for integration until cows came home. But to what end? Any/All definite integrals can be evaluated using numerical methods. Most integrals in application can not even be evaluated in elementary terms anyway.

So is integration like calligraphy?, where it is pretty to do/look at but a printer is the practical way to go.

Or are there other reasons behind learning evaluation of integrals in elementary terms beyond grading or personal joy?

PS : By practical application we don't mean oversimplified situations. Even the equation of pendulum is simplified so a nice answer for simple cases can be derived. In reality the change of potential energy with distance like calculations effecting satellite and probe launches to other planets can only be evaluated using numerical methods.

Answer 1

To some extent you are correct, but I would not go as far as comparing exact integration to calligraphy. There are enough practical situations where an exact solutions using integration by one of the numerous integration techniques actually can be computed. Moreover, numerical techniques have their problems. They could be costly, taking a long time to compute, and there is the issue of how accurate they are.

So, for integration it is best to know at least some integration techniques as well as some numerical methods. In this way you learn to appreciate the pros and cons of each method and you should be proficient enough to be able to use whichever is most suitable, or a combination of the two, when solving practical problems.

I do think though that there is far too much emphasis in the standard curricula on integration techniques. This is largely due to historical reasons. It used to be a common task of mathematicians to solve the integrals of, say, physicists. Today the physicists just use some software (e.g., Wolfram alpha) to solve their integrals and, if the integral is standard enough, the computer will usually give a much better answer than a mathematician would do.

It's a bit like matrices. To understand matrix multiplication you need to multiply at least a few $4\times 4$ matrices, but there is no point in becoming a grand master of multiplying matrices (though it used to be a valuable skill before computers showed up). So is it with integration. To understand what it is you need to compute at least a few integrals using substitution and integration by parts, and see some general tricks, but today there is little justification to the over-drilling of integration techniques in undergraduate courses.

Answer 2

"Most integrals in application can not even be evaluated in elementary terms anyway."

That is an overstatement. For example, polynomials occur in applications, and their integrals can be evaluated in elementary terms. Just to take a very simple example: The position of a falling object (constant acceleration) is given by a quadratic function of time, which is obtained by integration.

Answer 3

To me, numerical integration is the last solution I should consider because of its cost and because the problem of accuracy. Special functions coming from integration of complex terms have received a lot of attention and specific, taylor made, subroutines have been developed for thei accurate evaluation. Why do you think that so many libraries of subroutines have been developed ?

Answer 4

Most integrals in application can not even be evaluated in elementary terms anyway.

This statement should be considered carefully and not taken as a general truth. Everything depends on what you mean by "integrals in application."

Yes, integrals that arise in scientific/engineering practice are typically complicated and require numerical methods to evaluate.

On the other hand, one of the most important reasons for one to learn integration is to understand one of the mathematical pillars supporting the entire theory of physics from the 17th century onward. If you have ever taken a serious physics course (classical mechanics, for example) then you do see exactly solvable integrals in the derivation of many formulas or approximations. Students who do not know the basics of integration cannot progress very far in physics because it is absolutely necessary to understand the reason certain formulas should hold.

Of course, this is not at all unique to physics. Even though in practice engineers almost never compute an integral by hand, when they are learning their trade they must learn to do so because integration is a technique that is required to understand the theory behind the material in their engineering classes.

To summarize by analogy, integration techniques are like learning the alphabet. You need to do it to read the language of the sciences (e.g. engineering and physics), but once you do know how to read you can get by without remembering what letter comes after Q.