The Numerical Integration Lab

This lab introduces students to modeling, fitting functions to data, and integration of experimental data. The lab grew out of conversations with experimental physicists and biologists who commented that students could integrate elementary functions in closed form, but when faced with functions represented by sets of experimentally generated data points, the students did not indicate an understanding of the geometric ideas behind Riemann sums.

This lab first presents data representing rates of change of carbon-dioxide in aquatic systems [Beyers]. The integration of this data gives limnologists an idea about whether a lake or river is healthy, or whether it is susceptible to algae blooms. After integrating a set of given data, the students are asked to generate their own data: each team of students gets in a car and drives around town while recording their speed and mileage at a series of times. Upon returning to campus, each team numerically integrates their velocity data (using an interactive ``data integrator'' that is part of their lab) and compares it to the total distance the team traveled. This experiment sparks fascinating discussions about sampling frequency, the assumptions that go into estimating (modeling) functions between data points, and the realization that there is never an absolute answer to the question ``what is the integral of this set of data?''

The lab also introduces (but does not pursue) ideas such as cubic splines, Fourier polynomials, error estimation, and ``functions of best fit.'' The data integrator associated with this lab has been used by many experimentalists from outside the University and outside of mathematics because of its ease-of-use and versatility.

Next: The Beam Lab
Up: Introduction
Frederick J. Wicklin <>
Last modified: Fri Nov 29 12:28:11 1996