# Composite Numerical Integration

## Summary

Students will learn Composite Trapezoidal rule, Composite Simpson's rule, and Composite Midpoint Rule. Students are also asked to compare the results from these different methods.
Keywords: Composite Trapezoidal rule, Composite Simpson's rule, Composite Midpoint Rule

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## Learning Goals

Students are expected to learn a piecewise approach to numerical integration that uses the low-order Newton-Cotes formulas. Specifically, students learn Composite Trapezoidal rule, Composite Simpson's rule, and Composite Midpoint Rule. Students are also asked to compare the results from these different methods.

## Context for Use

The assignment was designed for upper level science majors who is taking Numerical Analysis course. Students should understand the mathematical reasoning behind the three different formulas, They should able to form their own M-files on the three different algorithms or able to change the given M-files to different problems from the assignment.

## Description and Teaching Materials

To complete this project, students should have access to MATLAB. Prior to this assignment, students should have discussed the Newton-Cotes formulas for three different cases: Trapezoidal rule, Simpson's rule and Midpoint rule. Students can discuss the disadvantages of using these rules over large integration intervals, specifically,on large interval, using Newton-Cotes formulas are not accurate, and

interpolation using high degree polynomial is unsuitable because of oscillatory nature of high degree polynomials.

Composite Trapezoidal rule (Acrobat (PDF) 36kB Oct2 17)
Composite Simpson's rule (Acrobat (PDF) 43kB Oct2 17)
Composite Midpoint Rule (Acrobat (PDF) 39kB Oct2 17)

## Teaching Notes and Tips

Students needs to have good grasp of the Newton-Cotes formulas before going to the composite numerical integration.

## Assessment

Students submit their final results including MATLAB codes for each problems and comparisons of the three different composite numerical integrations.

## References and Resources

Numerical Analysis, Richard l.Burden, J. Douglas Faires, Ninth Edition