Why learn integration techniques?

[maze]

I agree with DavidWhitbeck here.

 

[FrogPad]

I am going to try one very last time to state my points.

(1) Doing elementary integrals is not a routine aspect of physical science and engineering at the research level.

(2) However all of that tedious drilling in calculus does help students shore up their ability to think and reason algebraically.

(2) At any level in your education, you can find insight in either analytic solutions or numerical solutions.

(3) Analyzing physics problems and gaining insight does not necessarily imply a need for a complete solution.

(4) Challenging problems in research are solved numerically more often than not. But insight is still readily gained from their solutions.

(5) And I in no way suggested that numerical methods should be taught to the exclusion of anything else, thank you very much!

Alright I'm done.

Hey man, don't get all feisty here. I'm very glad you brought your argument on the table, because it was needed. I again will say, I agree that saying:

"Analytic solutions offer insight. Numerical solutions do not" ---> IS WRONG.

However, I want to remind you that this thread started with "why learn integration techniques". All of my arguments were from more of an educational standpoint. I misinterpreted you. I honestly thought, that you were saying that symbolic integration techniques should be disregarded. I didn't understand this, because I couldn't dream of how you would teach engineering classes... I mean you have to have a foundation, and to think algebraically before getting into numerical routines (in my opinion) is very important. Personally, I remember those toy problems very well and like to build up from there. So having an analytic solutions gives me an idea of a very simple (and NON-REAL) solution and offers tremendous insight. As an example, in Control Theory a very powerful example is looking at controlling the dynamics of a car via a basic cruise control scheme.

What's the first thing you do with a problem like that? Neglect. Neglect. Neglect. It becomes unreal very fast. However, when learning the ideas is it really that important to model wind with extremely complicated models? nahh... the very basic control ideas can be taught without resorting to numerical methods. So for the foundation, learning the analytic methods is very important (in my opinion). However, once you get past the elementary stuff, numerical methods pick up.

If you were arguing the points (1) - (5) that you maid I agree with you.However, I do not agree with this part of what you said: (well I do... but not completely).

If you can find an analytic solution you're done. Coming up with and using a numerical scheme involves (a) the functions involved satisfy all the properties needed for the algorithm to work, (b) the method needs to be not only convergent, but efficient, how do you find the balance? (c) how do you estimate the error? when do you stop the program, and why?

Finding the analytic solution is far from trivial (we are not just talking about integration techniques anymore right?).

I can't remember the exact quote but it was something to do with the Russians and the United States during the space-race times. It was something like, "when solving a challenging problem, the US throws more computers at it, while the Russians just invent new math".

I totally butchered the quote, but I hope the point remains. Obviously neither are trivial. Writing a numerical routine is complicated, and solving a problem analytically is also complicated. I just want to restate that solving analytically can be extremely and impossible.

 

[Ben Niehoff]

I was doing you a favor, that supports my argument, not yours. It cleanly debunks the silly image you constructed of a numerical analyst that goes problem --> blackbox --> answer. Coming up with an algorithm, implementing, and extracting results from the solution requires just as much thought, if not more, than using analytic methods. From the bottom of my heart, I thank you for making the point for me.

Nonsense. If you're going to split hairs over "methods" vs. "solutions", then a numerical solution IS a blackbox answer to a problem!

The method of writing a program to give an efficiently-convergent answer requires both extensive knowledge of how algorithms are implemented on computers (e.g., what sort of calculations are fast), and strong skills in traditional, pencil-and-paper mathematics, in order to take the mathematical methods and break them down into steps that can be efficiently calculated on a computer.

Remember the original question of this thread: "Why learn integration techniques?"

I'm pretty sure you are not actually arguing against the teaching of integration techniques anyway, so the rest of this is moot. No one could ever write a program to do integration without some prior knowledge of integration techniques, which is a point you've emphatically agreed to.

As to whether people in professional positions actually do symbolic calculus in their day-to-day duties, it really depends on the demands of the profession. Engineers, experimentalists, financial analysts, I'm sure use numerical solutions almost exclusively. But theorists and mathematicians need to work with analytical methods. They might use a symbolic package such as Mathematica to save themselves some tedium, but that is not strictly a "numerical" method anyway.