# ME205: Numerical Methods for Engineers

Unit 7: Numerical Integration   Integration may appear in a variety of numerical problems. For example, in regression or optimization problems, one may wish to minimize the integral of some measure of suboptimality or deviation. In these applications, one often has data chosen by experiment or some other constraint. At other times, one is free to choose where to sample data for integration. These options, to some extent, determine the algorithms available for numerical integration.*

*This unit reviews integral calculus, trapezoidal and quadratic integration, Romberg integration, and Gaussian quadrature.

This unit will take you approximately 11 hours to complete.

☐    Subunit 7.1: 2 hours

☐    Subunit 7.2: 2 hours

☐    Subunit 7.3: 3 hours

☐    Subunit 7.4: 4 hours

Unit7 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• Derive and apply the trapezoidal rule of integration.
• Derive and apply Simpson’s rule of integration.
• Distinguish Simpson’s method from the trapezoidal rule.
• Estimate errors in trapezoidal and Simpson integration.
• Derive and apply Romberg integration.
• Derive and apply the Gauss quadrature method of integration.

7.1 Review of Integral Calculus   - Web Media: University of South Florida: Holistic Numerical Methods Institute: Autar Kaw’s Lecture “Accumulation Function” Link: University of South Florida:  Holistic Numerical Methods Institute: Autar Kaw’s Lecture “Accumulation Function” (YouTube)

Instructions: View the entire video lecture (about 11 minutes).  Can you come up with a concise definition of accumulation function?

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Integral Calculus” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Integral Calculus” (PDF)

Instructions: Review the material in this chapter (24 pages).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Background of Integral Calculus” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Background of Integral Calculus” (PDF)

Instructions: Complete the entire multiple choice quiz.  Refer to resources in subunit 7.1 for any questions that pose difficulty.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

7.2 Trapezoidal and Simpson Integration   - Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Trapezoidal Method” Link: University of South Florida:  Holistic Numerical Methods Institute’s “Textbook Chapter of Trapezoidal Method” (PDF)

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Simpson’s 1/3 Rule” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter on Simpson’s 1/3 Rule” (PDF)

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Simpson’s 1/3 Rule” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Simpson’s 1/3 Rule” (PDF)

Instructions: Please complete the entire multiple choice quiz.  Refer to the resources in subunit 7.2 for any questions that pose difficulties.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Web Media: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Trapezoidal Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Trapezoidal Method” (PDF, DOC, HTML, FLASH)

Instructions: Under “Multiple Choice Test,” select the link for your preferred format (PDF, DOC, HTML, or FLASH) to download the quiz.  Please complete the entire multiple choice quiz.  Refer to the resources in subunit 7.2 for any questions that pose difficulties.  Please note that the solutions only appear in the Flash version of the quiz.

7.3 Romberg Integration   - Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Romberg Integration” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Romberg Integration” (YouTube)

Instructions: Under “Digital Audiovisual Lectures,” click on the YouTube link after each title to launch the videos.  Please view all 6 videos in their entirety (approximately 55 minutes).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Romberg Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Romberg Method” (PDF)

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Romberg Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Romberg Method” (PDF)

Instructions: Complete the entire multiple choice quiz.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Chemical Engineering Example of Romberg Method” Link: University of South Florida: Holistic Numerical Methods Institute’s “Chemical Engineering Example of Romberg Method” (PDF)

Instructions: Read the entire example, and try to reproduce the calculations as you read.

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

7.4 Gaussian Quadrature Rules   - Web Media: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Gauss Quadrature” Link: University of South Florida: Holistic Numerical Methods Institute’s Lectures on “Gauss Quadrature” (YouTube)

Instructions: Under “Digital Audiovisual Lectures,” click on the YouTube link after the title of each video.  Please view all 7 videos in their entirety (approximately 75 minutes).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Textbook Chapter of Gauss Quadrature” Link: University of South Florida:  Holistic Numerical Methods Institute’s “Textbook Chapter of Gauss Quadrature” (PDF)

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Assessment: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Gauss Quadrature” Link: University of South Florida: Holistic Numerical Methods Institute’s “Test Your Knowledge of Gauss Quadrature” (PDF)

Instructions: Attempt all questions on the multiple choice quiz.  You can find the answers here (PDF).

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

• Reading: University of South Florida: Holistic Numerical Methods Institute’s “Chemical Engineering Example of Gauss Quadrature” Link: University of South Florida: Holistic Numerical Methods Institute’s “Computer Engineering Example of Gauss Quadrature” (PDF)

Instructions: Follow the entire example, and try to reproduce the calculations as you read.

Terms of Use: The article above is released under a Creative Commons Attribution-Non-Commercial-Share-Alike License 3.0 (HTML).  It is attributed to the University of South Florida and the original version can be found here (HTML).

7.5 An Exercise in Numerical Integration   - Assessment: The Saylor Foundation’s “ME205: Unit 7.5 Exercise” Link: The Saylor Foundation’s “ME205: Unit 7.5 Exercise” (PDF)

Instructions: Please perform this exercise.  When you are done, check your work against The Saylor Foundation’s “ME304: Unit 7.5 Exercise Solution Guide" (PDF).  This exercise should require less than 3 hours to complete.