[Published in PRIMUS,
Volume VI, No. 4 (December 1996), 366-380]

ABSTRACT: This article explores the ways that a computer algebra system may be
used in conjunction with plotting software to allow students to investigate the
convergence properties of Riemann sums for functions of one or two variables in a
way that is not possible with just numerical calculations. Explicit formulas for
the Riemann sums suggest algebraic combinations that lead to the trapezoid and
Simpson's rules as a way to improve the rate of convergence.

KEYWORDS: Riemann sums, trapezoid rule, Simpson's rule, rates of convergence.

In many recent calculus texts (and some of the older ones, too!), the definite
integral is first thought of geometrically as area, and some form of the
Fundamental Theorem is often obtained through constructing and solving a
differential equation. Only later is a connection made with the analytic
definition of the integral as the limit of a Riemann sum. One of the main
stumbling blocks with using Riemann sums to define the definite integral for
beginning calculus students has been the difficulty in expressing the Riemann sum
in a closed form that permits the evaluation of the limit. Without this step we
have been forced to rely mainly on numerical calculations and animated pictures
of rectangles "filling" the region under the curve as more and more (thinner)
rectangles are drawn to convince our students that these sums actually do
converge. Computer algebra systems such as Derive, Mathematica, and Maple,
however, can easily simplify Riemann sums for all polynomials and for functions
of the form sin(mx), cos(mx), and exp(mx), thereby providing a wealth of examples involving the
standard functions of calculus that students can use to investigate the
convergence properties of these sums. In particular, we would like our students
to

understand the definition of a Riemann sum;

demonstrate that the
limit of a Riemann sum is the same for (almost) any choice of points from each
subinterval;

appreciate the different rates of convergence of the
approximating sums; and

investigate combinations of these Riemann sums that
lead to better approximations.

The examples in this paper illustrate some of the
investigations I use with students to achieve these goals.