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Numerical integration can be described as set of algorithms for calculating the numerical value of a definite integral. Definite integrals arise in many different areas and calculus is a tool

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Introduction

image00.png

Aim:

Numerical integration can be described as set of algorithms for calculating the numerical value of a definite integral. Definite integrals arise in many different areas and calculus is a tool for evaluating them; with numerous applications in science and engineering as well as mathematical analysis. However, calculus cannot always be applied; there are functions which do not have antiderivaties. One such example is

image32.png;this is an important function since it will be used in this coursework.

Using my knowledge of numerical integration I shall produce an approximation to the function image33.png ,seeing as it cannot be integrated. The area will be evaluated between the values of 0 and 1 (radians). The graph below gives a visual representation of the area which I shall be calculating.

Explanation:

It is often complicated to find the analytic solution to many differential equations. However, to our benefit there are many methods for finding the approximate solutions to differential equations. These methods are referred to as polynomials: the mid-point rule, trapezium rule and Simpson’s rule. Before explaining the methods in detail, we should note that all of these methods presented do not produce exact solutions, only approximate ones.  

Lissaman R. (2004), suggests that the midpoint rule uses rectangles to approximate the area underneath a curve. Below is a diagram which makes use of the mid-point rule:image41.png

In figure 1.0 five rectangles, each with the same width, are used to approximate the area under the graph of a function f(x) between x = 0 and x = 1.

The widths of the rectangles are donated h. The height of the first rectangle in the example is at the mid-point of the interval 0 to 0.2; which is 0.1 (represented by dashed line in figure 1.0).

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Middle

image06.png

image08.pngimage07.png

On the other hand, the trapezium rule is an underestimate and each gap is a quarter to the one on its left. Likewise, the trapezium rule is also a second order polynomial and only underestimates when the curve is convex.image11.pngimage10.pngimage09.pngimage12.pngimage06.pngimage06.png

image14.png

Notice, that the mid-point rule is a mirror image of the trapezium rule and vice versa. This is because the mid-point overestimates whilst the trapezium rule underestimates. Therefore, by using both methods we can confine the exact value between the two different polynomials. This means by obtaining values of Tn and Mn where n is a large number, we can find an accurate approximation to the integral image32.png as we know the exact value is between that of Tn and Mn.

Technology:

Most of this coursework will be carried out using Microsoft Excel; an electronic spreadsheet program used for organizing and manipulating data. The program is capable of working accurately up to 16 decimals places; this is also the amount I have chosen to work with in my calculations. Due to its manipulative ability Excel makes the handling of data very easy and saves an enormous amount of time. This is due to the fact that you don’t need to calculate everything; once the formula is entered and two consecutive calculations are complete, the cells can be dragged down and the answers required appear.

Excel is able to do this as it follows the formula and is judicious. The reason I chosen to use a program instead of a calculator is due to its ability to use 16 decimal places whereas a calculator can only give answers accurate to 9 decimal places.

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Conclusion

n+Tn)/3. The average is twice as close to Mn as it is to Tn. This can be justified by the differences in the errors associated with the two polynomials. This is explained below:

Estimate the area of I=image38.pngimage39.pngusing the trapezium rule and mid-point rule, only using 1 strip.

Trapezium rule:         Mid-point Rule:

 h= (4-0)/1= 4           h= (4-0)/1= 4

x

0

4

f(x)

0

16

x

2

f(x)

4

T1 = ½ * 4 * (f(0)+f(4)) = 32        M1= 4* (f(2)) = 16

The area of image39.pngcan be worked out from integration and is exactly 21 1/3. Therefore the absolute error in T1 is (32-21 1/3) = 10 2/3, whereas that in M1 is (21 1/3- 16) = 5 1/3. From this we can see that the absolute error in T1 is twice as much as that in M1.

image40.png

You can see from inspecting the errors in the trapezium sum and the midpoint sum that the midpoint sum is about twice as accurate as the trapezoidal sum and opposite in sign. This explains the weighting of the formula (2Mn+Tn)/3. By using the Simpson’s rule we will have cancellations of the errors and should thus get a much more accurate approximation.

The advantage of using the Simpson’s rule is that it’s a fourth- order polynomial. As mentioned before Simpson’s fits a parabola between successive triples of pairs. The absolute error is proportional to h4 so it is able to achieve a more accurate approximation to the curve,image42.png.

Absolute error ∝ h4 image37.png Absolute error = kh4

As a result the absolute error is Sn= kh4 whereas the absolute error in S2n= k(h/2)4. This means the errors reduces by a scale factor of 1/16 between each successive Sn values. In theory, the results I have obtained for Sn should be more accurate and reach the exact value in fewer calculations. This is seen to be the case by looking at the spreadsheet attached to this coursework.

image43.png

...read more.

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