Dependent on the amount of trapezia (n) used the general formula is:
2. The Midpoint Rule:
The Midpoint Rule divides the area underneath the curve into rectangles. We can then use the formula (where a, b are the 2 different sides of the rectangle) to estimate the area.
Dependent on the amount of rectangles (n) used the general formula is:
3. Simpson’s Rule
Simpson’s Rule is a weighted average of and . The normal average is exactly in the middle of and . But in this rules the average is twice as close to as it is to . This is justified by the differences in the errors associated with both rules.
The general formula is:
Simpson’s rule always requires an even number of strips.
4. Strategy
I have chosen this particular strategy because all three rules work well together – e.g.
- There is a connection between the Trapezium Rule and the Midpoint Rule which can be used to shorten calculations:
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Quick calculation of using and
- All three rules simplify the working out on the spreadsheet due to less difficult formulae
Formula Application
I used the programme “Microsoft Excel 2003” to produce the spreadsheet. To work with formulae in Excel is easier than using the calculator and saves a lot of time. It is more accurate as well, using values accurate to 15 decimal places.
On the following page I give a brief explanation about the calculations involved.
Error Analysis
Because all three rules are just estimates, there is always an error involved. In this section I will give an estimate of how much error there is in my solution of the problem.
Firstly I want to refer to my graph of my function on page 1. The section of the curve is concave that means that the Trapezium Rule gives an overestimate and the Midpoint Rule gives an underestimate. For that reason the estimates of the Trapezium Rule are decreasing and they will reach the solution from above. On the other side the Midpoint Rule estimates are increasing and they will reach the solution from below.
Error in the Trapezium Rule
In the Trapezium Rule the error is proportional to h2. We add a constant k so that:
In general, if is the absolute error in then there is a constant (k) so that:
Where h is the strip width corresponding to n strips.
So the Trapezium Rule with 2n strips has a strip width of , such that:
That shows that halving h or doubling n will reduce the error by a factor of 0.25.
Therefore the “error multiplier” is 0.25. The Trapezium Rule is a “second order method”.
Error in the Midpoint Rule
The error connected to the Midpoint Rule is the same as the error as in the Trapezium Rule.
That means that halving h or doubling n will reduce the error by a factor of 0.25.
Therefore the “error multiplier” is 0.25. The Midpoint Rule is a “second order method”.
Error in Simpson’s rule
The error in Simpson’s rule is proportional to h4. Again we add the constant k so that:
In general if is the absolute error in then there is a constant (k) so that:
Where h is the strip width corresponding to n strips.
So Simpson’s rule with 2n stripes has a strip width of , such that:
That shows that halving h or doubling n will reduce the error by a factor of 0.0625.
Therefore the “error multiplier” is 0.0625. The Trapezium Rule is a “fourth order method”.
Error in terms of differences and ratio of differences
The “error multiplier” mentioned in the part above is the same as the ratio of differences on the spreadsheet.
This can be proved by looking on the diagram below.
The distance between and is 4 times smaller than the distance between and , where X represents the actual solution. To get the next distance between and we multiply the “multiplier” 0.25 with the distance between and .
This development leads to improved solution by extrapolation, as stated in the “Formula Application” section.
Error in my solution
The error gets smaller the more strips are used as shown above. Therefore:
Interpretation
⇒
My solution to the integral is: 0.885909 (6 decimal places)
The solution refers to: T4096= 0.885909120684554
M4096= 0.885909049759261
S256= 0.885909075135303
The fact that the Trapezium Rule and the Midpoint Rule give the same solution till the sixth decimal place guaranties that my solution is valid. Simpson’s Rule, the weighted average, proves the solution as well.
Therefore: My solution is proved by 3 different rules.
I can say with guarantee that there is an error involved, because all three rules are just approximation rules. But I can say that my solution is not far away from the actual solution as the error for the rules with 4096 stripes is very small (actual value stated in the “Error Analysis” section).
