Why learn integration techniques?

[Gib Z]

Perhaps you all interpreted too much emphasis of the pen/paper bit. I don't personally like proving theorems with computers, like the 4 color theorem, but that's another argument. I was wanted to state that basically its *nice* to have things evaluated in exact terms, and in the most elementary form possible. Everyone knows how to simplify algebraic expressions and the like, we can state the same thing in many ways, but we want the simplest one to work with. Same one applies here. Thats all I wanted to say.

 

[Eidos]

@defennder: I suppose you are right. The four colour theorem does fall into that category.
What I meant to say was that computers cannot solve original problems all by themselves. Someone had to program in the routine used.

@maze: That was awesome! :D
Thanks for the laugh.

 

[jimvoit]

Hard Cold Logic

Here is an exerp from Wikipedia. "Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements is the earliest known systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) ."

The same question of "utility" has been directed to the teaching of Euclidean geometry in high school. I think the utiliterians prevailed on this one and computer assisted learing products such as The Geometers Sketchpad have prevailed to the exclusion of the hard cold logic necessary to follow a set of axioms to a correct conclusion.

I'm not an educator so maybe someone in the education community can amplify these comments.

 

[DavidWhitbeck]

The same question of "utility" has been directed to the teaching of Euclidean geometry in high school. I think the utiliterians prevailed on this one and computer assisted learing products such as The Geometers Sketchpad have prevailed to the exclusion of the hard cold logic necessary to follow a set of axioms to a correct conclusion.

I'm not an educator so maybe someone in the education community can amplify these comments.

That's terrible! High school geometry was what turned me onto math and science. I loved doing those proofs, you know where you had to do those constructions on one side, and list the propositions, axioms etc on the other side line by line to arrive at the logical conclusion that you wanted to show. I didn't love math before that class, but after I knew that I wanted to go into either physics or math. If it wasn't for that class, I might have ended up in the humanities instead!

Anyway that explains a lot. My students had no concept of logical argumentation, and even in calculus were taught to "prove" an identity by assuming it and working it through to 1=1. Sigh.

 

[FrogPad]

yes that was a mistake on my part.
but calculus classes are for engineers, why do they need to learn these techniques? I'm not an engineering student i don't know but i would guess their integrals are simple? even if they are why waste their time when everyone these days has access to some way of evaluating them numerically.
te integral and derivative are very important, it's the rigamarole i don't see the necessity of.

Analytic solutions offer insight. Numerical answers do not.

I would say, yeah... as a student, we deal with simple integrals. I mean, really, what else would we get? Would it be realistic to give a integral that requires some type of numerical method to solve it for? I have obviously had homework, questions, but not on a test.

As an example, one may use linear approximations to complicated expressions and then solve these simplified expressions. Often times these simplified expressions are valid enough, and they offer loads more insight into a problem than a sequence of numbers.edit:
Many times higher order terms are chopped out so things can be formed into nice algebraic solutions for a first order approximation. This gives someone a feel of the problem, and then when the accuracy is required we move onto using a nice numerical routine to include all the "hard" terms. As an example, one can reduce systems that have cross coupling terms as they could be negligible for a first order approximation. Another example is dealing with transistor models, the "real" models (which sim software such as Cadence uses) is way more complicated then the simple expressions engineers come to memorize. However, these simple expressions allow us to get an idea if the numbers being spit out of the computer are within a reasonable measure of accuracy.

Also, if we didn't teach basic methods how would you even test a numerical method you write on the computer? The first thing I do when writing a method from scratch is plug in a problem that I know the answer to.

 

[gel]

if most integrals aren't integrable to and can be evaluated numerically to any degree of accuracy why learn these esoteric techniques at all?

To answer the original question - there's lots of reasons. One is that computers can be very slow. They may seem fast enough at first, but if you want to do double integrals, triple integrals etc, then it soon becomes impossible.
If you can evaluate some of the integrals analytically, or simplify it in some other way, then the speedup can make things possible which would otherwise take much too long.

 

[symbolipoint]

... How about speed and efficiency of the calculations made by the computer?
FrogPad has the correct idea stated very neatly. An engineer who has an integral, maybe a complicated integral, may obtain benefit if he can transform the integral either into another form, equivalent expression, or into an algebraic formula. What is important is the "insight", of how the quantities are associated with each other.

 

[gmax137]

Solving a definite integral numerically rather than symbolically may be OK if what you're interested in is just the numerical value, but it's not everything. I was looking at the nearby thread on "difficult integral" (arctan(x))/(x*(x**2+1)). I decided to do it numerically using an excel spreadsheet and came up with a value (0.729...). So what? The first thind I did was to try to see if that value corresponded to something interesting (pi/4 or something like that). If it had (or even if the calculated value had been something else, say 0.75000...) then maybe I'd look harder for the "symbolic" solution. Somehow it is just more meaningful, or deeper, to know the symbolic result.

Of course, this doesn't answer the other question (why do the work if you have software that does the symbolic integration).

 

[matt grime]

I would like to throw a different position into the mix: the reason you learn these things is little to do with integrals really.

Why suppose that the sole reason to learn something is to mindlessly apply it? What does an engineer do? They solve (engineering) problems. How do they do this? They look into their metaphorical tool bag and decide which tool suits the job. If they don't have it, then they should try to work out what will solve the problem at hand, perhaps by adapting an existing method, or inventing a new one entirely. To do this they need education in how to estimate what the potential solution might be, as well as the ability to apply what they know, and to recognise when what they know isn't enough.

Sounds a lot like what a course on learning integration techniques teaches you. If you want a low brow analogy, think wax on wax off in Karate Kid.

 

[dirk_mec1]

I would like to throw a different position into the mix: the reason you learn these things is little to do with integrals really.

If you solve an engineering problem with a integral of (x^2+1) term in it, an arctan(x) pops up. If don't know the derative of this function you immediately start to look if you did something wrong, while in fact everything is okay!

Knowing the basics of solving integrals (to a certain level) gives the engineer the tools to check or interpretet the answers and that's what's this is about. If things get to difficult you resort to numerical methods.

Why suppose that the sole reason to learn something is to mindlessly apply it? What does an engineer do? They solve (engineering) problems. How do they do this? They look into their metaphorical tool bag and decide which tool suits the job. If they don't have it, then they should try to work out what will solve the problem at hand, perhaps by adapting an existing method, or inventing a new one entirely. To do this they need education in how to estimate what the potential solution might be, as well as the ability to apply what they know, and to recognise when what they know isn't enough.

Exactly my point.

 

[Cvan]

I'm an engineering undergraduate. Integration is an absolutely fundamental technique (not really a technique as much as a beautiful relationship) for solving problems dealing with engineering and physics on almost all levels. There is not a better way to put this than to compare the original question to: If there exists a library database online, why bother to read--since all the knowledge you could ever want you could just look up? That stresses a mentality that will drive modern civilization through the floor--a mentality that doesn't believe that understanding "why" things are the way they are is important.

Since you're looking for examples, consider a few of the following.
Thermodynamic heat transfer cycles (engines! motors! etc!)
Pressure, work, power,
Torque, kinematics, a whole slew of electromagnetic applications...the list goes on. Hell, there isn't much in an engineer's curriculum that having a fundamental understanding of math and physics DOESN'T explain (practically). That's probably why engineers take math and physics courses!

As for different kinds of engineers (I'm a mechanical with aerospace concentration for electives)--I know that my MET friends (mechanical engineering tech) are already done with math after getting through some basic multi variable, since the scope of the jobs they're looking at is more in the quality control department, being out on the field.

 

[DavidWhitbeck]

Analytic solutions offer insight. Numerical answers do not.

I strongly disagree with this. Let's do it case by case. Suppose that you wanted the integral of some function.

Case 1: You are a pure mathematician, you might be more interested in properties about the antiderivative that can be proved without actually evaluating it. In that case neither analytic nor numerical solutions are important.

Case 2: You are in applied math/physical science/engineering and are studying the properties of the function in the context of a toy problem. In that case you will gain intuition by examining and understanding the qualitative features of the function visually i.e. by a plot. Then either analytic or numerical solutions will fit the bill.

Case 3: You are in applied math/physical science/engineering and are solving a research problem. In that case it is not the formula that's useful, but how the antiderivative depends on a certain set of parameters. Evaluating the integral and obtaining these types of meaningful results are best done numerically.

Case 4: You are studying a textbook and they need to evaluate an integral as an intermediate step in finding an expression only valid under special circumstances not applicable in the real world, i.e. research. Alright you have me there.

The point is that if you fixate on Case 4 then you think that analytic trumps numerical always and that's almost never the case. Numerical solutions can offer insight, it depends on the problem. Blanket statements that say that numerical solutions do not offer insight or obviously wrong.

 

[Cvan]

If you don't substitute values into your problem, and solve from an analytical standpoint, you have solved an infinite number of problems, including the one you're looking at.

If you sub in early, you only solve one, and the relationship between your beginning criterion and output may not be as intuitive from a numerical standpoint.

 

[DavidWhitbeck]

If you don't substitute values into your problem, and solve from an analytical standpoint, you have solved an infinite number of problems, including the one you're looking at.

If you sub in early, you only solve one, and the relationship between your beginning criterion and output may not be as intuitive from a numerical standpoint.

That's misleading. If you write an algorithm to numerically compute something, it will be a function of the parameters you pass to it. That means that you do have something that solves an infinite number of problems. You just have to call it and evaluate it for each choice of parameter, but you have to do for the analytic solution as well! Provided that there is an analytic solution and you can find it easily.

 

[kts123]

I strongly disagree with this. Let's do it case by case. Suppose that you wanted the integral of some function.

Case 1: You are a pure mathematician, you might be more interested in properties about the antiderivative that can be proved without actually evaluating it. In that case neither analytic nor numerical solutions are important.

Case 2: You are in applied math/physical science/engineering and are studying the properties of the function in the context of a toy problem. In that case you will gain intuition by examining and understanding the qualitative features of the function visually i.e. by a plot. Then either analytic or numerical solutions will fit the bill.

Case 3: You are in applied math/physical science/engineering and are solving a research problem. In that case it is not the formula that's useful, but how the antiderivative depends on a certain set of parameters. Evaluating the integral and obtaining these types of meaningful results are best done numerically.

Case 4: You are studying a textbook and they need to evaluate an integral as an intermediate step in finding an expression only valid under special circumstances not applicable in the real world, i.e. research. Alright you have me there.

The point is that if you fixate on Case 4 then you think that analytic trumps numerical always and that's almost never the case. Numerical solutions can offer insight, it depends on the problem. Blanket statements that say that numerical solutions do not offer insight or obviously wrong.

Quantum Mechanics strongly disagrees with that. Most of what we know about QM has been given rise to by studying relationships between various known formula. General Relativity uses a heavy emphasis of differential geometry -- the concepts being used, of course, would be meaningless without understanding the Calculus. Most advanced mathematics must be viewed through the scope of integrals and derivatives; For example, even getting knee deep in complex analysis will throw you straight into evaluating integrals and derivatives (one nice example is being able to express complex numbers in terms of re^i^\theta, something which would make no sense if we didn't have calculus.)

Simply put, numerical solutions are impossible if you don't have an understanding of the mathematics at work -- to get the numerical solution we REQUIRE the understanding in the first place. At best your argument can be "someone solved it for me already."

 

[Eidos]

Quantum Mechanics strongly disagrees with that.

I'm afraid that Quantum Mechanics has no opinion on any matter whatsoever, it is a model of the world not a self-aware agent

I would most go with what Matt Grime has said, if we are speaking about this sort of thing in a strictly engineering context. I have noticed though, even with my own collegues (in undergrad) a reluctance to learn the theory of some of the maths we have been taught and focus solely on 'getting the right answer'.

The problem with that thinking though is that you are likely to apply some tool out of context and not knowing what to expect, arrive at a solution which is invalid. The nice thing though, engineering being practical by nature, is that you experiment and if the experiment is not what was predicted by theory to within a given degree of error, then you simply try again.

I think that having computers evaluate our monkey-maths work, it frees our time to concentrate on other parts of the design process. Much like machines do on factory floors, by automating a tedious job, you free up hands to do other useful work further up the production line

 

[DavidWhitbeck]

kts your post is a complete strawman. We are not talking about understanding calculus, we are talking about analytic vs numerical results.

Quantum Mechanics strongly disagrees with that. Most of what we know about QM has been given rise to by studying relationships between various known formula.

Sigh, quantum mechanics is founded on linear algebra in Hilbert spaces. Are you one of those people that thought it was always about solving the Schrödinger eqn. in the position representation? Hahaha!

General Relativity uses a heavy emphasis of differential geometry -- the concepts being used, of course, would be meaningless without understanding the Calculus.

And what does that have to do with analytic vs numerical results? That's a little off topic don't you think? Understanding calculus is not the same as analytic computation involving elementary functions. You must understand calculus to develop and use numerical algorithms as well, did you realize that?

Could you try to actually read all of the posts before replying next time?

 

[DavidWhitbeck]

Let's go over this for those just tuning in (so this doesn't happen again)--

dx, ice and mute are arguing about the importance of symbolic integration in practical engineering on page 1

gib, eidos, defennder, crosson are arguing about pen and paper vs computer results on page 2

crosson, gib, eidos are discussing what cas (such as mathematica) can't do that humans can on page 2

frogpad, symbolipoint, qmax, cvan and I, myself are discussing insight in analytic and numerical solutions on page 3

Matt, dirkmech, Cvan, Eidos are discussing calculus as a set of tools for engineers on pages 3 and 4.

There are also a couple of discussions that I didn't mention. The point is that this should help posters pick up on the context of the post and follow the right discussion. That's how a thread like this can work-- you really have multiple, independent discussions going on with different perspectives and goals.

 

[kts123]

Oops, that was a strawman. Ignore me, I shouldn't have been skimming.

 

[matt grime]

I'm not discussing calculus as a set of tools for engineers. I couldn't care less about the importance or otherwise of integration per se to engineers. I'm pointing out that you learn more in a mathematics class than just a technique to solve an integral, but learn a valuable lesson in how to actually think.

 

[FrogPad]

Blanket statements that say that numerical solutions do not offer insight or obviously wrong

I agree with you. I should not have said that numerical solutions do not offer insight. However, I think it was obvious that I was making a point... if someone doubts the need and utility of numerical methods, then we've got a problem.

I don't think we cannot underestimate the need and utility of integration techniques though. They support the basic foundation and seem to be useful as a tool for teaching engineering.

I wanted to speak on a few of your points individually:

Case 2:
Toy problems are important. Toy problems can capture the same complicated ideas as non-toy problems, but in a much easier to understand format. If you had a solution to a toy problem that was analytic would you use the numerical one?

Case 3:
Referring to:

...meaningful results are best done numerically.

You can't say that meaningful results are best done numerically in research. Obviously this really depends on the problem. This is just as much as a blanket statement as the one I made. But your point is there, and I can't say I disagree.

 

[DavidWhitbeck]

I'm not discussing calculus as a set of tools for engineers. I couldn't care less about the importance or otherwise of integration per se to engineers. I'm pointing out that you learn more in a mathematics class than just a technique to solve an integral, but learn a valuable lesson in how to actually think.

Sorry for misrepresenting you.

 

[epenguin]

Limiting to the question actually asked, I had hesitantly expressed the same doubt before.*
I suspect this will go the same way as whether you are allowed to use calculators in schools for arithmetic. Maybe if you have learned one way, you find it hard to imagine not doing that can be sound.

Isn't it true that most people never forget how to differentiate, but a year after a course on integration you will have forgotten it mostly unless you have used it fairly constantly?

Maybe for many courses in physics teachers could stop pretending they are solving equations, and just demonstrate the solutions?
At least I recommend to students if it is at all heavy, go straight to the solutions given, check that they are solutions, that may make it more obvious how they were solved.

And whatever the arguments, even if it is better to know than not to, you have to set that against the time and effort taken away from other things it would be desirable to know. I suspect that there are so many of these that in the long run this technique of no great cultural value will not be able to hold its place.


*"In a sense the only way to solve a differential equation or integration is to know the answer, or at least enough of its shape to be able to hammer it till it fits. I admit to subversive questionings about whether therefore it is a valid thing to teach or give so much emphasis on it and integrations, given that it will be used only by real professionals, and for understanding physics you only need those answers, which you can check."

 

[Ben Niehoff]

I'll throw in my two cents here...

I say that it is easier to gain insight about a problem from an analytical solution than from a numeric solution. When talking of simple, one-dimensional integrals, the reasons are not so apparent; you can, after all, plot a graph of the solution either way. However, my recent course on Jackson's electrodynamics makes a good example of the value of analytical solutions, because vector fields and potentials in 3-dimensional space are not so easy to visualize (and even harder in higher-dimensional spaces!).

The best example I remember from the class is the problem of a (positive) point charge in the region exterior to a (positively) charged, conducting sphere. What does the field look like? What is the behavior of the system when the point charge is in the regions A) very close to the surface of the sphere, B) near the sphere, but not extremely close, C) far away from the sphere, and D) in the limit at infinity?

With a numerical solution, the best you can do is plot values and draw a picture. If you know the right plots to make, you might learn something. But making the right plots takes a bit of guesswork. You can gain a much more general understanding by deriving the analytical formula and taking the appropriate limits. You would discover:

A) Very near the sphere, the point charge is actually attracted (despite being of the same sign), as though the sphere were an infinite, uncharged conducting plane. That is, the force felt is toward the sphere, and constant w.r.t. \alpha, where \alpha is a small compared to R, the radius of the sphere.

B) Just outside the attractive region, the point charge and the sphere repel with a force that goes as the inverse cube of the distance between them.

C) Farther away, the usual inverse square repulsion is felt.

D) At infinity, the force goes to zero.

These are all easy to derive with the formula. How would you get them with a numerical solution?

Good luck with A. You would probably find that there is a region of attractive force, but unless you guessed correctly to plot things in terms of \alpha, you would miss the valuable insight that any conducting surface "looks like" an infinite plane up close.

B would also throw you for a loop. How would you find the region of inverse-cube variation, except by choosing to do power law fitting on that portion of the graph? Certainly that's a waste of computer resources, when a few minutes on paper can get an exact solution.

C you could probably guess, if you know anything about electrostatics.

And for D, finding the limits at infinity are always tricky with numerical solutions. You can graph something that looks like it should go to zero at infinity. But you can't prove it goes to zero at infinity unless you use a formula and take limits.Overall, I would say the ability to take limits is the primary advantage of analytic solutions. It allows you to derive definite trends in the solution, valid in different regions, and you can get a lot of insight on the problem without actually having to calculate anything. Also, being able to see how the mathematics works, symbolically, is what allows you to make connections between different bodies of mathematics, and extend theories in a universal, logical way.

For example, how could you use numerical solutions to demonstrate the equivalence of Newtonian and Lagrangian mechanics? You could write some programs to implement each, and you would find that they gave the same numerical answers, but you could never prove, deductively, that they are the same; all you would have is empirical evidence from your programs. This is no way to conduct a logical proof. The same goes for demonstrating the equivalence of Schrödinger's, Heisenberg's, and the path integral formulations of quantum mechanics.

Not to mention the fact that in order to have numerical methods in the first place, someone has to write the code for them, and that person has to have an understanding of the underlying mathematics in order to implement it correctly and efficiently. These things don't just spring into being, fully-formed. And how could the numerical programs be checked for correctness? By comparing them to known analytical solutions, of course!If all you wish to do is engineering, then for the most part you can get away with numerical solutions. All you really need are correct answers. But if you want to do anything more theoretical, or get an understanding of the problem without having to look up tables of results, then there is value in learning how to do things analytically.

 

[DavidWhitbeck]

I'll throw in my two cents here...

I say that it is easier to gain insight about a problem from an analytical solution than from a numeric solution.

This is cute because you then proceeded to give an example where you examined qualitative features without reference to the complete analytic solution. You did not make your point with that example!

With a numerical solution, the best you can do is plot values and draw a picture. If you know the right plots to make, you might learn something. But making the right plots takes a bit of guesswork.

What guess work are you talking about? Plot |F| vs |r|. That's the only thing you were talking about in this problem.

These are all easy to derive with the formula.

That's overkill, you don't need the full equation. Everything you stated could be deduced from Gauss' Law without the need for a full analytic solution.

you would miss the valuable insight that any conducting surface "looks like" an infinite plane up close.

That's common sense, a surface is locally flat because it's smooth. That has nothing to do with the analytic solution.

B would also throw you for a loop. How would you find the region of inverse-cube variation, except by choosing to do power law fitting on that portion of the graph?

Well I have an idea... perhaps you could simply note that in the multipole expansion it would be the first term that doesn't vanish. Hey check that out, I didn't even know the formula, and I still saw the truth! Funny that.

And for D, finding the limits at infinity are always tricky with numerical solutions.

This property is again obvious from the de, you don't need the solution to uncover the asymptotes, in fact that is a standard elementary calculus problem to find asymptotes from a de without solving it.

Bottomline: You are confusing the ability to mathematically analyze a problem with finding an analytic solution. Nothing stops you from figuring out qualitative features and using that to find a solution either analytically or numerically. And a good physicist can figure out the qualitative features of the solution without needing to extract the full solution using either method.

 

[Ben Niehoff]

This is cute because you then proceeded to give an example where you examined qualitative features without reference to the complete analytic solution. You did not make your point with that example!

If you wouldn't selectively ignore the important parts of my argument, you would see that my example does indeed make my point.

Bottomline: You are confusing the ability to mathematically analyze a problem with finding an analytic solution. Nothing stops you from figuring out qualitative features and using that to find a solution either analytically or numerically. And a good physicist can figure out the qualitative features of the solution without needing to extract the full solution using either method.

Here you are agreeing with me that you need analytical methods to gain actual insight about the problem. The fastest way to analyze this particular problem is, in fact, to derive the solution using the method of images, and then take limits. Attempting to find facts such as (B) would be extremely difficult from Maxwell's equations alone, if you insisted on never actually solving them.

You also neglected to address my point about who writes the numerical solution algorithms in the first place, and how they are checked for correctness.

 

[DavidWhitbeck]

Here you are agreeing with me that you need analytical methods to gain actual insight about the problem.

So what? You were debating the value of analytic solutions vs numerical solutions. What, did you actually think that I would reply to your strawman fallacy? Let me requote the thesis of your argument to remind you--

I say that it is easier to gain insight about a problem from an analytical solution than from a numeric solution.

Now that we've taken care of that let's move right along.

You also neglected to address my point about who writes the numerical solution algorithms in the first place, and how they are checked for correctness.

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.

 

[FrogPad]

Coming up with an algorithm, implementing, and extracting results from the solution requires just as much thought, if not more, than using analytic methods

ahhhhh...?



Got a question for you?
How would you teach a course in E&M? Would you use only numerical methods?


I'm kind of getting lost in your arguments here. Are you saying that we should not teach symbolic integration techniques? Or are you saying something more like, in education we should teach these integration techniques, but for real world engineering problems it is pointless?

And you didn't answer my question about toy problems. If you have an analytic solution to a toy problem, would you use it? Or would you do something more along the lines of show the differential equation (as an example), and just present numerical "answers" to it?

 

[DavidWhitbeck]

ahhhhh...?

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?

Many times in research the problems that you are trying to solve will be sufficiently complex and pathological that simple methods like Runge-Kutta and Gaussian Quadrature would make for poor choices. It's not as if you can just copy and paste code from Numerical Recipes and call it a day.

Got a question for you?
How would you teach a course in E&M? Would you use only numerical methods?

Thanks, another strawman, wonderful. Look, no we were discussing solutions not methods. You and your cohorts were arguing that analytic solutions offer insight, and numerical solutions do not. I can quote you and others if I need to. I'm getting really tired of it though. I am also sick and tired of having my view misrepresented as the complete antithesis of what you said.

I'm kind of getting lost in your arguments here. Are you saying that we should not teach symbolic integration techniques?

No, and in fact in my very first post on this thread, I said that the real value in teaching differential and integration techniques is that it shores up students' ability to reason and think algebraically.

Or are you saying something more like, in education we should teach these integration techniques, but for real world engineering problems it is pointless?

Yes that is what I'm saying. The complexity of physics/chemistry/engineering research problems far exceed what you see in elementary calculus. You need to understand calculus for all of those fields, but you won't be solving textbook calculus problems on a regular basis.

And you didn't answer my question about toy problems. If you have an analytic solution to a toy problem, would you use it? Or would you do something more along the lines of show the differential equation (as an example), and just present numerical "answers" to it?

I wouldn't necessarily do either. And if you carefully read my reply to Ben, you will understand why. Let me give you an example from when I am teaching physics. A common mechanics problem is to identify the terminal velocity given free fall in the presence of a drag force that varies as some function of velocity. Now, I know perfectly well the complete analytic solution to the problem. Do I use it? Do I show it to my students? No. Why? You don't need it, and it doesn't offer physical insight that is not present in Newton's 2nd Law. It was the same with Ben's example. You did not actually need the solution to deduce the properties that he mentioned. Now before you say this, because God knows you will... I'm not saying that you never have to solve the equations you write down.

The entire point of one of my earlier posts was that one should not confuse the method with the solution. You and Ben are treating them as interchangable, and that is why you are so clearly confused and so poorly misrepresenting my posts. Analyzing the physics of a problem is not necessarily the same as solving an equation.

 

[DavidWhitbeck]

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.