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).