# A Thought Experiment: Integrating Rates of Change

Suppose you experimentally measure the rate at which some quantity is
changing. Maybe you have ten measurements, maybe you have a thousand
measurements. In any case, you want to determine the total
change in the quantity that occurred over the time interval when you
collected data. How might this total change be approximated?
For example, suppose you work for the Environmental
Protection Agency (EPA) and are investigating a possible violation of
the Clean Air Act. As part of your job you travel to a factory to
measure the rate at which the factory is belching filth into the air.

By flashing your badge and threatening the factory manager with
imprisonment, you convince the manager to allow you to climb to the
top of the factory's smokestack four times during the work shift.
Each time you reach the top of the smokestack, you measure the rate at
which soot is emitted for one minute. After one minute, burning eyes
and a hacking cough force you to climb back down to the ground level
to recover from the experience.

### Figure 1: Collecting experimental data.

The chart below summarizes the data you collected:
Time Measured rate of soot
production (kg/hour)
------ -----------------------
8:00 2
10:00 3
1:00 4
5:00 1

From this data you need to know: how much soot (in kilograms) did the
factory spew into the air during the day?
Recall that derivatives are rates of change. Furthermore, the derivative
and integral are inverse operations, so
given the **rate** at which soot enters the air, we can
obtain the **amount** of soot entering the air by
integrating our rate data. A problem arrives, however, when we
realize that we do not know the underlying "pollution-rate function" that
determines the rate at which soot enters the air! All we have are
some data points for this pollution-rate function!

Thus the purpose of this lab: how can we integrate functions that we
don't know explicitly? More specifically, how can we "integrate"
experimental data?

**Next:**Models of Experimental Data

**Return to:**Introduction

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

Last modified: Fri Jan 5 09:51:08 1996