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Integration - definite integral

We now introduce the definite integral by considering an example where integration is used to find an amount which is represented by the area under a curve.

Story 1

A leaking pipe is discovered.
The rate at which the water is leaking is measured and is found to be 4 L/h.
After 1 hour, the rate of leaking is again measured and it is still 4 L/h.
After 2 hours, the rate is still 4 L/h.
After 3 hours, the rate is still 4 L/h.

Estimate the amount of water that will escape from the pipe in the 24 hours following the last measurement.

That's easy. Assume that the rate of leaking remains constant at 4 L/h.

estimate of amount = 24 h x 4 L/h = 96 L

Story 2

A leaking pipe is discovered.
The rate at which the water is leaking is measured and is found to be 4 L/h.
After 1 hour, the rate of leaking is again measured and it is 4.2 L/h.
After 2 hours, the rate is 4.4 L/h.
After 3 hours, the rate is 4.6 L/h.

Assuming that this increasing pattern continues, estimate the amount of water that will escape from the pipe in the 24 hours following the last measurement.

In both stories, the amount of water is the area under the graph of the rate of change plotted against time.

Using integration to find the amount in story 2:

Let litres be the amount escaped in hours since the leak was first discovered i.e. the area under the graph starting at t=0.

subst.

and

subst.

estimate of volume that will escape from the pipe in the 24 hours following the last measurement
= 180.9 - 12.9 L
= 168 L

This result can be verified by direct calculation.

The calculation in story 2 can be presented more concisely using special notation known as the definite integral. The values 3 and 27 are respectively the lower limit and the upper limit of the definite integral.

This example also demonstrates the concept that the amount of change of a quantity is given by the area under the graph of the rate of change of that quantity. For example, the change in displacement of a body is given by the area under the velocity-time graph (velocity being the rate of change of displacement with respect to time).

Use of TI-83

Definite integrals can be calculated using the MATH menu.
For most functions, this is a numerical approximation rather than an exact result.

The corresponding calculation can be performed on the graph using the CALC menu (details).

Definite integrals / areas can be negative.

In this example, the positive area above the x-axis is matched by a negative area below the x-axis.

Find a cubic function that would fit this situation.

Definite integrals can be calculated using the MATH menu. For most functions, this is a numerical approximation rather than an exact result.
The corresponding calculation can be performed on the graph using the CALC menu (details).