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Riemann Sums and Area Approximations

This page and activity authored by James Rutledge, St. Petersburg College.
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This material is replicated on a number of sites as part of the SERC Pedagogic Service Project

Summary

After covering the standard course material on area under a curve, Riemann sums and numerical integration, Calculus I students are given a write-pair-share activity that directs them to predict the best area approximation methods for each of several different functions. Afterwards, the instructor employs a Web-based applet that visually displays each method and provides the corresponding numerical approximations.

Learning Goals

To enable students to:
  • develop their understanding of area approximation methods
  • recognize the usefulness of numerical integration
  • gain an understanding of the limit process in obtaining area
  • apply their knowledge and understanding in determining the best approximation methods for given functions

Context for Use

This activity is appropriate for a Calculus I class and provides an opportunity for students to exercise their analytical skills with regard to area approximations. It was designed for a standard classroom but may be carried out in large lecture classes as well.

The activity requires approximately 30 minutes, including the instructor's presentation of the Web-based applet and class discussion.

Description and Teaching Materials

Activity description:
  • Students are given a write-pair-share activity worksheet (Rich Text File 34kB Jan22 07) involving area approximation methods in which they are asked to sketch graphs and representative rectangles, trapezoids and parabolas for each of several functions and then predict the most accurate approximation methods: Riemann left endpoints, Riemann midpoints, Riemann right endpoints, Trapezoidal Rule or Simpson's Rule.

  • Afterwards, the instructor presents an interactive Java applet that demonstrates the five area approximation methods so that students can verify their analyses. The number of intervals may be increased to provide a sense of the limit process in finding area as well as to promote discussion about the number of intervals needed for desired accuracy

Teaching Notes and Tips

This activity leads students to take a closer look at the various types of Riemann sums for area under a curve as well as the numerical approximations involving trapezoids and parabolas. The series of quick sketches that is required gives students a more tangible sense of the methods and associated approximation errors.

The accuracies of the estimation involved with f(x)=sin(x), [0, 6.28318], are somewhat misleading due to the symmetric nature of the function over its domain; in actuality, there is quite a bit of error involved with some of the methods. This can be pointed out to students and then verified when examining the next function and its domain, f(x)=sin(x), [0, 3.14159]

To get a visual display of the area (and error involved) and to lead toward a discussion of the limit process as well as toward a discussion of how many intervals might be required to achieve a desired accuracy, the number of intervals in the applet may be increased to 20, 40, 80, 100, 1000, etc.

Assessment

Assessment is optional; the intent is to develop students' understanding of area approximation methods.

References and Resources

Numerical Integration Rules
For a given rational function, this applet allows the user to enter the number of intervals and select an area approximation method. The output includes a visual display of the graph and area approximation along with numerical values for the approximate area and the absolute value of the error. Doubling the number of intervals is an option and the use of this option clearly reveals the reduction in error with larger numbers of intervals.
MERLOT description of the Numerical Integration Rules resource.
Direct link to Numerical Integration Rules site.


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