4.2 The Definite Integral

Summary

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. Applets Riemann Sums Videos See short videos of worked problems for this section. Quiz Take a quiz. 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.

Resources on the Web Information on Newton Biographical data from St. Andrew's University's Web site Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics" Information on Leibniz Biographical data from St. Andrew's University's Web site Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics" Calculus Applications Project Intermath Riemann Sums Visual Calculus

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4.1 Modeling Accumulations

Table of Contents

4.3 Properties of the Definite Integral

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