1. MECN 3500 Inter - Bayamon Lecture 10101010 Numerical Methods for Engineering MECN 3500 Professor: Dr. Omar E. Meza Castillo omeza@bayamon.inter.edu http://www.bc.inter.edu/facultad/omeza Department of Mechanical Engineering Inter American University of Puerto Rico Bayamon Campus

2. Lecture 10 MECN 3500 Inter - Bayamon 2 Tentative Lectures Schedule TopicLecture Mathematical Modeling and Engineering Problem Solving 1 Introduction to Matlab 2 Numerical Error 3 Root Finding 4-5-6 System of Linear Equations 7-8 Least Square Curve Fitting 9 Numerical Integration 10 Ordinary Differential Equations

3. Lecture 10 MECN 3500 Inter - Bayamon Newton-Cotes Integration Formulas Numerical Integration 3

4. Lecture 10 MECN 3500 Inter - Bayamon  To solve numerical problems and appreciate their applications for engineering problem solving. 4 Course Objectives

5. Lecture 10 MECN 3500 Inter - Bayamon Introduction

6. Lecture 10 MECN 3500 Inter - Bayamon They are based on the strategy of replacing a complicated function or tabulated data with an approximating function that is easier to integrate: where f n (x) is a polynomial of degree n. f1(x)f1(x) f2(x)f2(x)

7. Lecture 10 MECN 3500 Inter - Bayamon Piecewise functions can be used also to approximate the integral. 3 piecewise linear functions to approximate f(x) between a and b.

8. Lecture 10 MECN 3500 Inter - Bayamon Two forms of the Newton-Cotes formulas: Closed Forms: the data points at the beginning and end of the limits of integration are known. Open Forms: integration limits extend beyond the range of the data.

9. Lecture 10 MECN 3500 Inter - Bayamon The Trapezoidal Rule The integral is approximated by a line:

10. Lecture 10 MECN 3500 Inter - Bayamon Statement: Use the trapezoidal rule to estimate Example 21.1 Solution:

11. Lecture 10 MECN 3500 Inter - Bayamon One way to improve the accuracy of the trapezoidal rule is to divide the integration interval from a to b into a number of segments and apply the method to each segment. The areas of individual segments can then be added to yield the integral for the entire interval. The Multiple-Application Trapezoidal Rule

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14. Lecture 10 MECN 3500 Inter - Bayamon The total integral is Substituting the trapezoidal rule for each integral: Grouping terms:

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16. Lecture 10 MECN 3500 Inter - Bayamon Statement: Use the multiple-application trapezoidal rule for n = 2 to estimate Example 21.1 Solution:

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18. Lecture 10 MECN 3500 Inter - Bayamon Computer Algorithms for the Trapezoidal Rule

19. Lecture 10 MECN 3500 Inter - Bayamon More accurate estimate of an integral is obtained if a high-order polynomial is used to connect the points. The formulas that result from taking the integrals under such polynomials are called Simpson’s rules. Simpson’s Rules

20. Lecture 10 MECN 3500 Inter - Bayamon This rule results when a second-order interpolating polynomial is used. After integration, Simpson’s 1/3 Rule

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22. Lecture 10 MECN 3500 Inter - Bayamon Statement: Single Application of Simpson’s 1/3 Rule From a=0 to b=0.8. recall that the exact integral is 1.640533 Which is approximately 5 times more accurate than for a single application of the trapezoidal rule (Example 21.1) Example 21.4 Solution:

23. Lecture 10 MECN 3500 Inter - Bayamon This rule results when a third-order interpolating polynomial is used. Simpson’s 8/3 Rule

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25. Lecture 10 MECN 3500 Inter - Bayamon Statement: Single Application of Simpson’s 3/8 rule to integrate From a=0 to b=0.8. Simpson’s rule 3/8 requires four equally spaced points: Example 21.6 Solution:

26. Lecture 10 MECN 3500 Inter - Bayamon Statement: Estimate the cross section area of the stream. Case Studies

27. Lecture 10 MECN 3500 Inter - Bayamon Consider this example

28. Lecture 10 MECN 3500 Inter - Bayamon Trapezoidal rule (h = 4): Trapezoidal rule (h = 2):

29. Lecture 10 MECN 3500 Inter - Bayamon Integration with Matlab Use quad for functions. Software

30. Lecture 10 MECN 3500 Inter - Bayamon Homework8  www.bc.inter.edu/facultad/omeza www.bc.inter.edu/facultad/omeza Omar E. Meza Castillo Ph.D. 30