1. EE3561_Unit 7Al-Dhaifallah1435 1 EE 3561 : Computational Methods Unit 7 Numerical Integration Dr. Mujahed AlDhaifallah ( Term 342)

2. EE3561_Unit 7Al-Dhaifallah14352 Lecture 19 Introduction to Numerical Integration  Definitions  Upper and Lower Sums  Trapezoid Method  Examples

3. EE3561_Unit 7Al-Dhaifallah14353 Integration Indefinite Integrals Indefinite Integrals of a function are functions that differ from each other by a constant. Definite Integrals Definite Integrals are numbers.

4. EE3561_Unit 7Al-Dhaifallah14354 Fundamental Theorem of Calculus

5. EE3561_Unit 7Al-Dhaifallah14355 The Area Under the Curve One interpretation of the definite integral is Integral = area under the curve ab f(x)

6. EE3561_Unit 7Al-Dhaifallah14356 Riemann Sums

7. EE3561_Unit 7Al-Dhaifallah14357 Numerical Integration Methods Numerical integration Methods Covered in this course  Upper and Lower Sums  Newton-Cotes Methods:  Trapezoid Rule  Romberg Method

8. EE3561_Unit 7Al-Dhaifallah14358 Upper and Lower Sums ab f(x) The interval is divided into subintervals

9. EE3561_Unit 7Al-Dhaifallah14359 Upper and Lower Sums ab f(x)

10. EE3561_Unit 7Al-Dhaifallah143510 Example

11. EE3561_Unit 7Al-Dhaifallah143511 Example

12. EE3561_Unit 7Al-Dhaifallah143512 Upper and Lower Sums Estimates based on Upper and Lower Sums are easy to obtain for monotonic functions (always increasing or always decreasing). For non-monotonic functions, finding maximum and minimum of the function can be difficult and other methods can be more attractive.

13. EE3561_Unit 7Al-Dhaifallah143513 Newton-Cotes Methods  In Newton-Cote Methods, the function is approximated by a polynomial of order n  Computing the integral of a polynomial is easy.

14. EE3561_Unit 7Al-Dhaifallah143514 Newton-Cotes Methods Trapezoid Method ( First Order Polynomial are used ) Simpson 1/3 Rule ( Second Order Polynomial are used ),

15. EE3561_Unit 7Al-Dhaifallah143515 Trapezoid Method f(x)

16. EE3561_Unit 7Al-Dhaifallah143516 Trapezoid Method Derivation-One interval

17. EE3561_Unit 7Al-Dhaifallah143517 Trapezoid Method f(x)

18. EE3561_Unit 7Al-Dhaifallah143518 Trapezoid Method Multiple Application Rule ab f(x) x

19. EE3561_Unit 7Al-Dhaifallah143519 Trapezoid Method General Formula and special case

20. EE3561_Unit 7Al-Dhaifallah143520 Example Given a tabulated values of the velocity of an object. Obtain an estimate of the distance traveled in the interval [0,3]. Time (s)0.01.02.03.0 Velocity (m/s)0.0101214 Distance = integral of the velocity

21. EE3561_Unit 7Al-Dhaifallah143521 Example Time (s)0.01.02.03.0 Velocity (m/s) 0.0101214

22. EE3561_Unit 7Al-Dhaifallah143522 Estimating the Error For Trapezoid method

23. EE3561_Unit 7Al-Dhaifallah143523 Error in estimating the integral Theorem

24. EE3561_Unit 7Al-Dhaifallah143524 Example

25. EE3561_Unit 7Al-Dhaifallah143525 Example x1.01.52.02.53.0 f(x)2.13.23.42.82.7

26. EE3561_Unit 7Al-Dhaifallah143526 Example x1.01.52.02.53.0 f(x)2.13.23.42.82.7

27. EE3561_Unit 7Al-Dhaifallah143527 Lecture 21 Romberg Method  Motivation  Derivation of Romberg Method  Romberg Method  Example  When to stop?

28. EE3561_Unit 7Al-Dhaifallah143528 Motivation for Romberg Method  Trapezoid formula with an interval h gives error of the order O(h 2 )  We can combine two Trapezoid estimates with intervals h and h/2 to get a better estimate.

29. EE3561_Unit 7Al-Dhaifallah143529 Romberg Method First column is obtained using Trapezoid Method R(0,0) R(1,0)R(1,1) R(2,0)R(2,1)R(2,2) R(3,0)R(3,1)R(3,2)R(3,3) The other elements are obtained using the Romberg Method

30. EE3561_Unit 7Al-Dhaifallah143530 First Column Recursive Trapezoid Method f(x)

31. EE3561_Unit 7Al-Dhaifallah143531 Recursive Trapezoid Method f(x) Based on previous estimate Based on new point

32. EE3561_Unit 7Al-Dhaifallah143532 Recursive Trapezoid Method f(x) Based on previous estimate Based on new points

33. EE3561_Unit 7Al-Dhaifallah143533 Recursive Trapezoid Method Formulas

34. EE3561_Unit 7Al-Dhaifallah143534 Recursive Trapezoid Method

35. EE3561_Unit 7Al-Dhaifallah143535 Derivation of Romberg Method

36. EE3561_Unit 7Al-Dhaifallah143536 Romberg Method R(0,0) R(1,0)R(1,1) R(2,0)R(2,1)R(2,2) R(3,0)R(3,1)R(3,2)R(3,3)

37. EE3561_Unit 7Al-Dhaifallah143537 Property of Romberg Method R(0,0) R(1,0)R(1,1) R(2,0)R(2,1)R(2,2) R(3,0)R(3,1)R(3,2)R(3,3) Error Level

38. EE3561_Unit 7Al-Dhaifallah143538 Example 1 0.5 3/81/3

39. EE3561_Unit 7Al-Dhaifallah143539 Example 1 cont. 0.5 3/81/3 11/321/3

40. EE3561_Unit 7Al-Dhaifallah143540 When do we stop?