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Topic: Re: Riemann Definition
Replies: 2   Last Post: Sep 21, 2004 11:25 AM

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 Ladnor Geissinger Posts: 55 Registered: 12/4/04
Re: Riemann Definition
Posted: Sep 21, 2004 11:25 AM

I could not let this recent comment on the role of Riemann sums pass by
without a few skeptical words.

>>Gupta: I believe that Riemann Sums are essential to introduction of
definite integrals and to all the applications that emanate from there. The
concepts of areas, volumes, moments, surface areas, arc lengths are, and
should be, based on the concepts of partitioning and partial sums. The fact
remains that many simple problems can be modelled in terms of definite
integrals that just can't be evaluated using the fundamental theorem-- here
the basic definition of the integral *must* be used to get any useful
value.
>>

Most calculus books proclaim as theme some variant of "calculus is the
mathematics of change". They then proceed to do differential calculus with
a mostly dynamic viewpoint. However, when integral calculus comes along,
there is an obvious switch to statics, and the impression is left that we
have to do Riemann sums to get solutions, or even to discuss
satisfactorily, area, volume, density, moment, etc. problems. This leads
to a lot of handwaving about definitions and existence of integrals and
fuss about the "fundamental theorem(s) of calculus". Students soon notice
that after the initial flurry, the author only resorts to Riemann sums as
"justification" for claiming a certain integral as the solution of some
problem, that is, setting up the problem seems to use Riemann sums but then
the author goes back to finding antiderivatives to actually solve problems.
All the (1-variable) integral calculus problems in (at least standard)
texts can be solved without ever mentioning Riemann sums; just recast the
problem in dynamic terms using the cumulative area, volume, mass, moment,
etc functions. Let a moving line sweep across the region and consider the
function which accumulates the desired quantity (e.g. Area(x) is the area
under the curve from "start" to x.). In every case it's simpler to
convince the student (and yourself) what the derivate of this function is
(use a single slice x to x+h) than to use Riemann sum approximation (many
slices). This just emphasizes the point some people make anyway, that the
subject is really about differential equations.

Following the DE theme suggests replacing the fuss and hand waving about
definition of integral and fund. theorems with a single fact: existence
and uniqueness of a continuous solution of y'=f(x) , y(start)=0, for
piecewise continuous f . (Keep the assumptions on f simple - maybe right
continuous everywhere and left hand limits exist at points of
discontinuity.) This result will seem at least as obvious to a student as
statements about Riemann integrals. As Strang suggests, lets make the
most of their experience with cars and other projectiles. Actually, a
second obvious fact will also be useful: if f is nonnegative, then so is
the solution. Of course this doesn't eliminate Riemann sums completely,
it just means that they appear as a way to get numerical approximations to
quantities, not as "justification" of the existence of the quantities or of
formulas for their computation.