1. Numerical Integration

2. Introduction to Numerical Integration
Definitions
Upper and Lower Sums
Trapezoid Method (Newton-Cotes Methods)

3. Integration Indefinite Integrals Definite Integrals
Indefinite Integrals of a function are functions that differ from each other by a constant.
Definite Integrals
Definite Integrals are numbers.

4. Fundamental Theorem of Calculus

5. The Area Under the Curve
One interpretation of the definite integral is:
Integral = area under the curve
f(x) a b

6. Upper and Lower Sums The interval is divided into subintervals. f(x) a

7. Upper and Lower Sums
f(x) a b
CISE301_Topic7

8. Example

9. Example

10. 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.

11. 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.

12. Newton-Cotes Methods
Trapezoid Method (First Order Polynomials are used)
Simpson 1/3 Rule (Second Order Polynomials are used)

13. Trapezoid Method Derivation-One Interval Multiple Application Rule
Estimating the Error
Recursive Trapezoid Method

14. Trapezoid Method
f(x)

15. Trapezoid Method Derivation-One Interval

16. Trapezoid Method
f(x)

17. Trapezoid Method Multiple Application Rule
f(x) x a b

18. Trapezoid Method General Formula and Special Case

19. 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)
0.0
1.0
2.0
3.0
Velocity (m/s) 10 12 14
Distance = integral of the velocity

20. Example 1
Time (s)
0.0
1.0
2.0
3.0
Velocity (m/s) 10 12 14

21. Error in estimating the integral Theorem
CISE301_Topic7

22. Estimating the Error For Trapezoid Method

23. Example

24. Example x
1.0
1.5
2.0
2.5
3.0
f(x)
2.1
3.2
3.4
2.8
2.7

25. Example x
1.0
1.5
2.0
2.5
3.0
f(x)
2.1
3.2
3.4
2.8
2.7

26. Recursive Trapezoid Method
f(x)

27. Recursive Trapezoid Method
f(x)
Based on previous estimate
Based on new point

28. Recursive Trapezoid Method
f(x)
Based on previous estimate
Based on new points

29. Recursive Trapezoid Method Formulas

30. Recursive Trapezoid Method

31. Example on Recursive Trapezoidh
R(n,0)
(b-a)=/2
(/4)[sin(0) + sin(/2)]= 1
(b-a)/2=/4
R(0,0)/2 + (/4) sin(/4) = 2
(b-a)/4=/8
R(1,0)/2 + (/8)[sin(/8)+sin(3/8)] = 3
(b-a)/8=/16
R(2,0)/2 + (/16)[sin(/16)+sin(3/16)+sin(5/16)+ sin(7/16)] =
Estimated Error = |R(3,0) – R(2,0)| =

32. Advantages of Recursive Trapezoid
Gives the same answer as the standard Trapezoid method.
Makes use of the available information to reduce the computation time.
Useful if the number of iterations is not known in advance.