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Topic: DE's vs Riemann sums
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Paul Zorn

Posts: 325
Registered: 12/6/04
DE's vs Riemann sums
Posted: Dec 16, 1993 1:48 PM
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In ref to Riemann (or other approximating) sums:

I agree with Lad Geissinger and others that the DE theme deserves
more emphasis than it usually gets in introductory calculus. Lad's
criticism that

> after the initial flurry [over Riemann sums],
> 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

is also on the mark. However, I think that the problem Lad correctly
identifies has less to do with intrinsic merits or demerits (mathematical
and pedagogical) of approximating sums than with too narrowly-conceived
problems and exercises. As Lad says:

> 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 ...

That's true for problems posed ``continuously'' (e.g., with data
given by explicit formulas) but not necessarily for problems with, say,
tabular data. (Estimate distance given discrete speed data; estimate
volume of a solid of revolution given discrete circumference data (what
you'd get with a tape measure ... ); estimate area of a plane lamina (a lake,
say) given discrete ``width'' data.) I'm convinced that doing such
problems---ALONG WITH more traditional continuous-data problems---helps
students understand BOTH the discrete and the continous better.

Lad also rightly points out that the ``static'' nature of
Riemann sums can be jarring after all the earlier talk about
dynamics. True, but that doesn't have to be bad: continuous
AND discrete points of view are important. What IS bad (as Lad
observes) is if the discrete point of view appears only as
window dressing, soon to go back into mental storage (i.e., oblivion).

Paul Zorn

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