12. Simpson's rule
The inaccuracy of the trapezium rule
In Core 2 the trapezium rule was introduced in order to enable the candidate to give estimates of definite integrals which they were not able to integrate accurately (or at all). It is relatively easy to see the ways in which the trapezium rule is inaccurate, merely drawing a diagram can show how the line of the trapezium will leave a large area which is not accounted for, and the larger the width of the trapezium, the greater the inaccuracy.
A geometric basis for Simpson's rule
Linear approximations of curves will generally be inaccurate as the curve will change in gradient, and the line does not. The only solution is to use shorter lines, however, in the context of the trapezium rule this is inaccurate.
A parabola can be used instead of a straight line. The parabola has a quadratic equation describing it, whose three coefficients can be defined to make the parabola fit the curve in question.
Suppose there is a section of curve, from "a", to "b", whose y-coordinates are 
, and 
respectively.
Let there be a midpoint, and the approximation parabola to have an equation 
.
From the midpoint of the region allow there to be "h" units either way (and define an origin at the mid point), such that the area under this approximation parabola is:
.
Consider the value of this integral:
![\left[ \frac{p}{3}x^{3} + \frac{q}{2}x^{2} + rx \right] ^{h} _{-h} =](http://www.thestudentroom.co.uk/latexrender/pictures/32/3212cc18b17e7b88beef5ce7d67310a9.png "\left[ \frac{p}{3}x^{3} + \frac{q}{2}x^{2} + rx \right] ^{h} _{-h} =")
.
There is no dependence on "q" in this.
Hence:
Hence:
.
Hence:
.
Hence, for a single interval Simpson's rule is defined:
Where 
.
This implies that there must be an even number of sub-intervals, and that intervals are equal in width.
Simpson's rule: general form
As with the trapezium rule, Simpson's rule benefits from more intervals, and therefore it is common to use a number of equal intervals.
Divide the interval from "b" to "a" into "2n" equal intervals of width "h".
To shorten the writing (as with the trapezium rule), allow the following to be true: 
.
Hence, using a method of simply repeating Simpson's rule:
Hence:
.
Due to the nature of Simpson's rule, the method of looking at the graph to see whether an underestimation or an overestimation has been made (as done with the trapezium rule) cannot be replicated.
As there is a lot of calculation involved it is important to be organised; using a table is helpful:
.
Make a note of "h" also.
