# Numerical Integration:

## Accumulating Rates of Change

The fundamental theorem of calculus tells us that if we know the rate of change of some quantity, then adding up (or integrating) the rate of change over some interval will give the total change in that quantity over the same interval. For example, if a car is moving along a straight line and we know the speed of the car as a function of time, it is possible to determine the total change in the car's position over some time interval. But what if we don't know a formula for the car's velocity, but we only have measured its velocity at certain instants of time? Is it possible to "integrate" this discrete data in order to estimate the change in the car's position? If so, how?

In this lab we learn to model functions that produce experimental data. By integrating the model, we approximate the (true) integral of the underlying (unknown) function. First, we integrate pre-collected data concerning the rate at which carbon-dioide is produced in an aquatic environment. Then we create, collect, and analyze data concerning the relationship between velocity and position.

## Outline

This lab is long. But some parts can be done independently of others. After completing the first three sections below, you can move on to the section on CO2 concentrations, or directly to the section on automobile velocities.
Next:A Thought Experiment

A portion of this lab is based on a problem appearing in the Harvard Consortium Calculus book, Hughes-Hallet, et al, 1994, p. 174

Support for the Curriculum Initiative Project at the University of Minnesota has been provided by a grant from the National Science Foundation (DUE 9456095) and by the Geometry Center.

The Geometry Center Calculus Development Team
Last modified: Fri Jan 5 11:19:51 1996