
Re: What Is an Integral?
Posted:
Feb 4, 1998 2:06 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: > >> > The approach in beginning calculus (at least in my class) was to >> >derive the formula for surface area using the surface area of >> >increasingly small cone frustums and passing to the limit. >> >> How do you know that this limit actually equals the surface area in >> question? As previously stated, many books simply present this as a >> matter of intuitive plausibility. > >I believe that no more is possible. This is the definition of the surface >area.
My point is that surface area is not some new concept that one can define in an arbitrary way. We have an intuitive nonformal understanding of surface area, and when you define it formally, you are under an obligation to show that your definition agrees with our intuitive understanding. Hopefully you will be able to show that any other definition which agrees with the intuitive concept must be logically equivalent to the one you have given.
In my opinion, the correct way to do this is to state axioms for surface area that everyone will agree are intuitively selfevident. Then show that these axioms uniquely characterize the concept and that your definition satisfies these axioms.
According to what Herman says, this is a formidible problem, and one that I personally are not willing to read up on. But I object to the way that many calculus books (and not only calculus!) take the attitude that one can make things true by declaring them as a matter of definition.
This is, in my opinion, teaching students sloppy thinking.
> Of course you can go on to prove that the surface area has the >additive property, or you could start from the additive property and >derive the limit form. I think this is more difficult.
 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

