1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 29 Numerical Integration

2. Motivation AREA BETWEEN a AND b

3. Think as Engineers!

4. In Summary INTERPOLATE

5. In Summary Newton-Cotes Formulas Replace a complicated function or tabulated data with an approximating function that is easy to integrate

6. In Summary Also by piecewise approximation

7. Closed/Open Forms CLOSEDOPEN

8. Trapezoidal Rule Linear Interpolation

9. Trapezoidal Rule

11. Trapezoidal Rule Multiple Application

13. xa=x o x1x1 x2x2 …x n-1 b=x n f(x)f(x 0 )f(x 1 )f(x 2 )f(x n-1 )f(x n )

14. Simpson’s 1/3 Rule Quadratic Interpolation

15. Simpson’s 1/3 Rule

16. xa=x o x1x1 x2x2 …x n-1 b=x n f(x)f(x 0 )f(x 1 )f(x 2 )f(x n-1 )f(x n )

17. Simpson’s 3/8 Rule Cubic Interpolation

18. Simpson’s 3/8 Rule

19. Gauss Quadrature x1x1 x2x2

20. General Case Gauss Method calculates pairs of wi, xi for the Integration limits -1,1 For Other Integration Limits Use Transformation

21. Gauss Quadrature For x g =-1, x=a For x g =1, x=b

22. Gauss Quadrature

24. PointsWeighting Factors wi Function Arguments Error 2W0=1.0X0=-0.577350269 F (4) (  ) W1=1.0X1= 0.577350269 3W0=0.5555556X0=-0.77459669 F (6) (  ) W1=0.8888888X0=0.0 W0=0.5555556X0=0.77459669

25. Gaussian Points PointsWeighting Factors wi Function Arguments Error 4W 0 =0.3478548X0=-0.861136312 F (8) (  ) W 1 =0.6521452X1=-339981044 W 2 =0.6521452X2=- 339981044 W 3 =0.3478548X3=0.861136312

26. Gaussian Quadrature Not a good method if function is not available