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.