# Integration of Nonelementary Functions

#### Summary

The project is for students in a Multivariable Calculus class to use two different methods to numerically integrate a nonelementary function to a desired accuracy. The outcome of the project is a report describing the project.

## Learning Goals

By the end of the project, students should be able to

- write simple programs in MATLAB,

- write a technical paper describing the results of a project, and

- use their knowledge of MATLAB to solve scientific and mathematical problems in other classes.

- write simple programs in MATLAB,

- write a technical paper describing the results of a project, and

- use their knowledge of MATLAB to solve scientific and mathematical problems in other classes.

## Context for Use

This project is designed for students in a Multivariable Calculus class who have studied Taylor series, and have seen Simpson's Rule in an Integral Calculus class. The project is designed for students with no prior experience using MATLAB. The free MATLAB Onramp course, which takes about 2 hours to complete, is used as a homework assignment to get students familiar with using MATLAB. Students spend one hour of class time becoming familiar with the project and calculating the error bounds when using Simpson's Rule and Taylor series prior to meeting for a one hour computer lab.

## Description and Teaching Materials

The project allows students to determine the accuracy of using Simpson's Rule and Taylor polynomials to evaluate the nonelementary integral, . The project is divided into five parts, with detailed instructions to guide students through each part, and to show them the steps that are usually followed to solve a computational problem. The different parts of the project are

I. Introduction to MATLAB – Students take the MATLAB Onramp course as part of a homework assignment.

II. Determine Error Bounds – Analytically determine the number of intervals to use with Simpson's Rule and the number of terms needed in a Taylor polynomial to compute the integral with an error less than 0.0001.

III. Writing/Modifying Code – Describes the different sections of a program, how to create an algorithm, and a copy of MATLAB code to implement Simpson's Rule is provided that will need to be modified to solve the specific integral used for this project.

IV. Using MATLAB – Describes the steps to follow when using MATLAB on a computer.

V. Report – The format for the written report is provided along with what is expected in each section.

Student Handout for Integrating Nonelementary Functions Project (Acrobat (PDF) 515kB Nov7 19)

Solutions to Integrating Nonelementary Functions (Acrobat (PDF) 466kB Nov7 19)

Rubric for written report (Acrobat (PDF) 414kB Nov7 19)

Sample Student Report (Acrobat (PDF) 1.6MB Nov7 19)

I. Introduction to MATLAB – Students take the MATLAB Onramp course as part of a homework assignment.

II. Determine Error Bounds – Analytically determine the number of intervals to use with Simpson's Rule and the number of terms needed in a Taylor polynomial to compute the integral with an error less than 0.0001.

III. Writing/Modifying Code – Describes the different sections of a program, how to create an algorithm, and a copy of MATLAB code to implement Simpson's Rule is provided that will need to be modified to solve the specific integral used for this project.

IV. Using MATLAB – Describes the steps to follow when using MATLAB on a computer.

V. Report – The format for the written report is provided along with what is expected in each section.

Student Handout for Integrating Nonelementary Functions Project (Acrobat (PDF) 515kB Nov7 19)

Solutions to Integrating Nonelementary Functions (Acrobat (PDF) 466kB Nov7 19)

Rubric for written report (Acrobat (PDF) 414kB Nov7 19)

Sample Student Report (Acrobat (PDF) 1.6MB Nov7 19)

## Teaching Notes and Tips

Part I Introduction to MATLAB

The MATLAB Onramp course is a good way to get students used to working in MATLAB without having to spend class time going over the basics of MATLAB in class.

Part II Determine Error Bounds

This was the most difficult section for my students. Many students did not understand how to use the error bound formula for Simpson's Rule to calculate the number of intervals needed. It is probably a topic that was not stressed in their Integral Calculus class.

When finding an error bound for Simpson's Rule, some students had trouble taking the fourth derivative of and finding its maximum value on the interval [0, 1]. They didn't think of factoring each derivative to make it times a polynomial.

When the assignment was first used, students were asked to find the error bound for a Taylor series using Taylor's Inequality. It states that if for then the remainder of the series satisfies the inequality

for

Students really had trouble using this error bound, so when the project was revised, the way the error bound is calculated for the Taylor series was changed to use the Alternating Series Estimation Theorem.

Part III Writing/Modifying Code

Students benefit from working together to understand the code for implementing Simpson's Rule, and discussing what needs to be changed to evaluate the integral for this assignment.

Part IV Using MATLAB

Before coming to the computer lab, students need to read through the whole section to understand how they will be using MATLAB, and they should create two tables in a WORD document to their results.

Part V Report

Two students working together on the report usually produces a better report, and it gives students the chance give and receive peer feedback.

The MATLAB Onramp course is a good way to get students used to working in MATLAB without having to spend class time going over the basics of MATLAB in class.

Part II Determine Error Bounds

This was the most difficult section for my students. Many students did not understand how to use the error bound formula for Simpson's Rule to calculate the number of intervals needed. It is probably a topic that was not stressed in their Integral Calculus class.

When finding an error bound for Simpson's Rule, some students had trouble taking the fourth derivative of and finding its maximum value on the interval [0, 1]. They didn't think of factoring each derivative to make it times a polynomial.

When the assignment was first used, students were asked to find the error bound for a Taylor series using Taylor's Inequality. It states that if for then the remainder of the series satisfies the inequality

for

Students really had trouble using this error bound, so when the project was revised, the way the error bound is calculated for the Taylor series was changed to use the Alternating Series Estimation Theorem.

Part III Writing/Modifying Code

Students benefit from working together to understand the code for implementing Simpson's Rule, and discussing what needs to be changed to evaluate the integral for this assignment.

Part IV Using MATLAB

Before coming to the computer lab, students need to read through the whole section to understand how they will be using MATLAB, and they should create two tables in a WORD document to their results.

Part V Report

Two students working together on the report usually produces a better report, and it gives students the chance give and receive peer feedback.

## Assessment

The students will write a report to summarize their findings, along with a copy of the code used to evaluate the integral. A rubric is provided to assess the report.

## References and Resources

1. Nonelementary Integral, Wikipedia – Gives definition of nonelementary integrals.

https://en.wikipedia.org/wiki/Nonelementary_integral

2. Fresnel integral spiral – Shows diagram of Euler spiral.

https://en.wikipedia.org/wiki/Fresnel_integral

3. The Clothoid – Describes use of Euler spiral to connect sections of road or track going in different directions

https://pwayblog.com/2016/07/03/the-clothoid/

4. Normal Distribution – Common application for the integral being evaluated

https://en.wikipedia.org/wiki/Normal_distribution

5. Stewart, James, Calculus Early Transcendentals, 7th ed., Brooks/Cole Cengage Learning, 2012 – Reference book with material on Simpson's Rule, and Taylor polynomials.

https://en.wikipedia.org/wiki/Nonelementary_integral

2. Fresnel integral spiral – Shows diagram of Euler spiral.

https://en.wikipedia.org/wiki/Fresnel_integral

3. The Clothoid – Describes use of Euler spiral to connect sections of road or track going in different directions

https://pwayblog.com/2016/07/03/the-clothoid/

4. Normal Distribution – Common application for the integral being evaluated

https://en.wikipedia.org/wiki/Normal_distribution

5. Stewart, James, Calculus Early Transcendentals, 7th ed., Brooks/Cole Cengage Learning, 2012 – Reference book with material on Simpson's Rule, and Taylor polynomials.