Simpson’s 3/8 rule

Simpson’s 38 rule is a method for approximating a definite integral by evaluating the integrand at finitely many points. The formal rule is given by


where x1=x0+h, x2=x0+2h, x3=x0+3h.

Simpson’s 38 rule is the third Newton-Cotes quadrature formula. It has degree of precision 3. This means it is exact for polynomialsPlanetmathPlanetmath of degree less than or equal to three. Simpson’s 38 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,

0πsin(x)𝑑xπ6[sin(0)+4sin(π2)+sin(π)]= 2.094

However, Simpson’s 38 rule does slightly better.

0πsin(x)𝑑x(38)π3[sin(0)+3sin(π3)+3sin(2π3)+sin(π)]= 2.040
Title Simpson’s 3/8 rule
Canonical name Simpsons38Rule
Date of creation 2013-03-22 13:40:56
Last modified on 2013-03-22 13:40:56
Owner Daume (40)
Last modified by Daume (40)
Numerical id 11
Author Daume (40)
Entry type Definition
Classification msc 41A05
Classification msc 41A55