Simpson’s 3/8 rule
where , , .
Simpson’s rule is the third Newton-Cotes quadrature formula. It has degree of precision 3. This means it is exact for polynomials of degree less than or equal to three. Simpson’s rule is an improvement to the traditional Simpson’s rule. The extra function evaluation gives a slightly more accurate approximation . We can see this with an example.
Using the fundamental theorem of the calculus, one shows
In this case Simpson’s rule gives,
However, Simpson’s rule does slightly better.