Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: What Is an Integral?
Replies: 48   Last Post: Feb 5, 1998 3:47 PM

 Messages: [ Previous | Next ]
 Lee Lady Posts: 24 Registered: 12/12/04
Re: What Is an Integral?
Posted: Feb 5, 1998 3:21 PM

In article <T.Moore-0302981000180001@130.123.97.36>,
Terry Moore <T.Moore@massey.ac.nz> 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 limit-of-sums 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 Step-function 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
Step-function 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 step-function 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

Date Subject Author
1/9/98 Darrell Ryan
1/9/98 Michael R.
1/9/98 Nat Silver
1/10/98 feldmann@bsi.fr
1/10/98 Darrell Ryan
1/11/98 Herman Rubin
1/31/98 Terry Moore
2/1/98 Michael Joyce
2/1/98 Herman Rubin
2/2/98 Herman Rubin
2/5/98 Herman Rubin
2/2/98 Herman Rubin
2/2/98 Terry Moore
2/5/98 Herman Rubin
1/11/98 altavoz
1/11/98 Darrell Ryan
1/12/98 altavoz
1/12/98 Darrell Ryan
1/12/98 altavoz
1/10/98 Darrell Ryan
1/11/98 altavoz
1/11/98 Brieuc Seydoux
1/11/98 Nat Silver
1/11/98 Darrell Ryan
1/11/98 Nat Silver
1/12/98 Darrell Ryan
1/12/98 Nat Silver
1/12/98 Darrell Ryan
1/12/98 Wilbert Dijkhof
1/13/98 Horst Kraemer
1/13/98 Darrell Ryan