1. 1 Numerical Integration Dr. Asaf Varol avarol@mix.wvu.edu

2. 2 Numerical Integration Numerical integration is a primary tool used for definite integrals that cannot be solved analytically. A numerical integration rule has the form we investigate several basic quadrature formulas that use function values at equally spaced points; these methods are known as Newton-Cotes formulas. There are two types of Newton –Cotes formulas, depending on whether or not the function values at the ends of the interval of integration are used. The trapezoid and Simpson rules are examples of “closed” formulas, in which the endpoint values are used. The midpoint rule is the simplest example of an “open” formula, in which the endpoints are not used [2].

3. 3 Newton-Cotes Closed Formulas Trapezoid Rule One of the simplest ways to approximate the area under a curve is to approximate the curve by a straight line. The trapezoid rule approximates the curve by the straight line that passes through the points [a, f(a) and b, f(b)], the two ends of the interval of interest. We have x0=a, x1=b, and h=b-a, and then

4. 4 Example

5. 5 Matlab Program

6. 6 Diagram

7. 7 Newton-Cotes Closed Formulas Simpson’s Rule

8. 8 Newton-Cotes Closed Formulas Simpson’s Rule (Cont’d)

9. 9 Matlab Program

10. 10 Diagram

11. 11 Newton-Cotes Closed Formulas Midpoint Rule

12. 12 Figure Figure given on the right side compares the actual value of the area with that found by using the midpoint rule. The area given by the integral S (hatched) and the approximation using the midpoint rule (shaded) [2].

13. 13 Matlab Program

14. 14 Diagram

15. 15 Gaussian quadrature

16. 16 Gaussian quadrature

17. 17 Example

18. 18 End of Chapter 5

19. 19 References Celik, Ismail, B., “Introductory Numerical Methods for Engineering Applications”, Ararat Books & Publishing, LCC., Morgantown, 2001 Fausett, Laurene, V. “Numerical Methods, Algorithms and Applications”, Prentice Hall, 2003 by Pearson Education, Inc., Upper Saddle River, NJ 07458 Rao, Singiresu, S., “Applied Numerical Methods for Engineers and Scientists, 2002 Prentice Hall, Upper Saddle River, NJ 07458 Mathews, John, H.; Fink, Kurtis, D., “Numerical Methods Using MATLAB” Fourth Edition, 2004 Prentice Hall, Upper Saddle River, NJ 07458 Varol, A., “Sayisal Analiz (Numerical Analysis), in Turkish, Course notes, Firat University, 2001