
Re: What Is an Integral?
Posted:
Feb 5, 1998 3:21 PM


In article <T.Moore0302981000180001@130.123.97.36>, Terry Moore <T.Moore@massey.ac.nz> wrote: >In article <6b2kf6$dnj@news.Hawaii.Edu>, lady@Hawaii.Edu (Lee Lady) wrote: >SNIP< > >I applaud what you are trying to do. I really should look at your web >page. I suspect you didn't quite mean that you are deriving the integral >from the area under a curve. What you have said here indicates an >approach somewhat more like the Daniel integral, in which we define >an integral in terms of the additive property, plus a normalisation that >defines the integral of a constant function.
As I've tried to make clear in other postings, the intention in my notes <Http://www.Math.Hawaii.Edu/~lee/calculus/Integrals.pdf> is not to develop some alternative theory of the integral, but rather to show how existing theory can be applied to considerably simplify the treatment of most common elementary applications of integration.
The approach I'm using was indeed inspired by the ideas of Daniell/ Darboux/Bourbaki. (And thanks for informing me by email of the correct spelling of Daniell; and thanks again to Herman for mentionning Darboux.)
I do encourage students to think of the integral as being the area under a curve, but that's the least important part of what's in my notes. Mathematically, of course, thinking of the integral as area is irrelevant. One still has to rigorously define area, and then one is back to the limit of sums. However thinking of the integral as the area under a curve gives students something tangible (or perhaps I should say visual) to hold in mind and thus, in my opinion, helps overcome one of the main problems that students have with the integral, namely the abstractness of the limitofsums construction.
However they still learn about the limit of sums  just in a more tangible form. For when one asks the question of why the area under the graph of a velocity function should equal the distance traveled by the corresponding object, the explanation involves thinking of the area as being made up of tiny vertical strips (or, more precisely, seeing it as a limit of areas made up of vertical strips).
In the terminology I use in my notes, the relationship between velocity and distance has the Stepfunction Approximation Property. The notes actually devote quite a few pages to helping students understand the plausibility of the idea of approximating functions by step functions. This is illustrated both graphically and with concrete numerical calculations.
What is different in my approach is that I focus students' attention on the question of convergence  in my terminology, whether the Stepfunction Approximation Principle applies to a given relationship or not. In most calculus books, the attention is focused on the process of actually constructing step function approximations for a given application and the question of convergence is discussed briefly or, in many cases, simply taken for granted.
In fact, once one is convinced that it's legitimate in a particular application to use stepfunction approximations, it's quite unnecessary to actually construct these step functions. All the pretty pictures in calculus books of volumes of revolution being approximated by a sequence of disks or shells are quite unnecessary. All that one needs is to find a formula in the form of an integral which gives the correct result in the case of constant functions.
 Trying to understand learning by studying schooling is rather like trying to understand sexuality by studying bordellos.  Mary Catherine Bateson, Peripheral Visions lady@Hawaii.Edu

