Simpson's Rules are a set of methods for approximating integrals, named after Thomas Simpson. It is generally considered more accurate than using the Trapezoidal Rule. The most basic of the rules is the 1/3 rule.
Simpson's 1/3 Rule is (note this method is an approximation, not exact):
Some functions cannot be accurately approximated by Simpson's 1/3 Rule, in these cases using the following composite rule one can break up the integral into an even number of segments and preform the method on these individual segments to increase the accuracy.
Where
Simpson's 3/8 Rule, also sometimes called Simpson's second rule, is generally more accurate then the 1/3 rule. [1]
Given an integral, of interval
Simpson's 3/8 Rule is (note this method is an approximation, not exact):
Like the 1/3 Rule the 3/8 Rule can also be broken up into intervals in order to increase accuracy. The composite version of the 3/8 Rule requires that an the total number of intervals be a multiple of 3.
Where
If written out it would look like the following:
In German and some other languages, it is named after Johannes Kepler who derived it in 1615 after seeing it used for wine barrels (barrel rule, Keplersche Fassregel).
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