1. APPROXIMATE INTEGRATION
PROGRAMME 22
APPROXIMATE INTEGRATION

2. Introduction
Approximate integration
Proof of Simpson’s rule

3. Introduction
Approximate integration
Proof of Simpson’s rule

4. Introduction
By integrating by parts, it is easily shown that:
The result obtained is exact. However, as soon as a numerical value for e is entered into the right-hand side then the result becomes inexact because the value of e used would only be accurate to a given number of decimal places.
In this sense the numerical result is an approximation and only accurate up to a given number of decimal places

5. Introduction
For example, the table shows the different results for different values of e:
The more accurate the value of e chosen then the more accurate the final numerical result

6. Introduction
In the approximate integration that follows the level of accuracy is one level of accuracy less.
The approximation methods used are approximations to the process of integration.

7. Introduction
Approximate integration
Proof of Simpson’s rule

8. Introduction
Approximate integration
Proof of Simpson’s rule

9. Approximate integration
By series
By Simpson’s rule

10. Approximate integration
By series
We cannot evaluate the integral:
and obtain an exact value. Instead an approximation to the process of integration is made by substituting for ex, the Maclaurin series expansion of ex:

11. Approximate integration
By series
To give:
Accuracy is then increased by taking more and more terms of the series into consideration

12. Approximate integration
By series
Care must be taken to ensure that the substituted series inside the integral does converge for the range of values of the variable of integration.
For example, to evaluate:
It must be noted that the binomial expansion:
diverges for 2 ≤ x ≤ 4 which is the range of the variable of integration.

13. Approximate integration
By series
Instead, the integral is converted to become:
Where the substituted series now converges for 2 ≤ x ≤ 4 .

14. Approximate integration
By Simpson’s rule
The definite integral can be related to the area beneath the curve y = f (x):
Simpson’s rule is an approximation
to the area beneath the curve and so
in that sense is an approximation
to the process of integration.

15. Approximate integration
By Simpson’s rule
To find the area under the curve y = f (x), above the x-axis and between the values x = a and x = b:
Divide the figure into any even number (n) of equal width strips
Number and measure each ordinate: y1, y2, y3, . . ., yn+1

16. Approximate integration
By Simpson’s rule
(c) The area A of the figure is then given by:
where:

17. Introduction
Approximate integration
Proof of Simpson’s rule

18. Introduction
Approximate integration
Proof of Simpson’s rule

19. Proof of Simpson’s rule
Divide the area shown into an even number of strips (2n) each of width s. let the ordinates be y1, y2, y3, . . ., y2n+1. Then:

20. Proof of Simpson’s rule
Let the curve through A, B, C be approximated by y = a + bx + cx2 so that:
from which can be found that:

21. Proof of Simpson’s rule
Let A1 be the area of the first pair of strips, then:

22. Proof of Simpson’s rule
Similarly:

23. Proof of Simpson’s rule
Therefore, since:
then:

24. Learning outcomes
Recognize when an integral cannot be evaluated directly
Approximate integrals using series expansions
Use Simpson’s rule to approximate the area beneath a curve