APPROXIMATE INTEGRATION PROGRAMME 22 APPROXIMATE INTEGRATION

Introduction Approximate integration Proof of Simpson’s rule

Introduction Approximate integration Proof of Simpson’s rule

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

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

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.

Introduction Approximate integration Proof of Simpson’s rule

Introduction Approximate integration Proof of Simpson’s rule

Approximate integration By series By Simpson’s rule

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:

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

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.

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

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.

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

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

Introduction Approximate integration Proof of Simpson’s rule

Introduction Approximate integration Proof of Simpson’s rule

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:

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:

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

Proof of Simpson’s rule Similarly:

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

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