## 4.2 The Definite Integral

### Summary

The area problem is described. A general procedure based on the method of accumulations is described for finding the area under a curve and above an interval. The definite integral is defined and introduced.

By the end of your studying, you should know:

• How to find Upper and Lower Riemann sums.
• How to use sigma notation.
• How to find Left and Right Riemann sums.
• How to define the definite integral of a function over an interval.
• The theorem that if a function is Riemann integrable on an interval, the definite integral equals a limit of Riemann sums (specific wording is in textbook).
• The theorem that every continuous function is Riemann integrable.
• Consequences of the last two results for approximating definite integrals of continuous functions.

On-screen applet instructions: Use the pull-down menu to choose the number of subintervals and hence rectangles.

### Examples

Write the following sum in sigma notation:

Consider the function

Is it possible to find the area between this function and the x axis?

Calculate

using the limits of Riemann sums.

Riemann Sums

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 4.2 The Definite Integral (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel