1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 30 Numerical Integration & Differentiation

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

3. In Summary Also by piecewise approximation

4. Closed/Open Forms CLOSEDOPEN

5. Trapezoidal Rule Linear Interpolation

6. Trapezoidal Rule Multiple Application

8. 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 )

9. Simpson’s 1/3 Rule Quadratic Interpolation

10. Simpson’s 3/8 Rule Cubic Interpolation

11. Gauss Quadrature x1x1 x2x2

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

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

14. Gauss Quadrature

16. 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=0.8888888X1=0.0 W2=0.5555556X2=0.77459669

17. 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

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

19. Fig 23.1 FORWARD FINITE DIFFERENCE

20. Fig 23.2 BACKWARD FINITE DIFFERENCE

21. Fig 23.3 CENTERED FINITE DIFFERENCE

22. Data with Errors