# Simpson's Rule

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## Simpson's rule

[′sim·sənz ‚rül]*a,b*] is approximated by

*h*[ƒ(

*a*) + 4ƒ(

*g*+

*h*) + ƒ(

*b*)]/3, where

*h*= (

*b*-

*a*)/2; this is the area under a parabola which coincides with the graph of ƒ at the abscissas

*a*,

*a*+

*h*, and

*b*.

## Simpson’s Rule

a formula for approximating definite integrals. In has the form

where *h* = (*b* - *a*)/2*n* and *f _{i}* =

*f*(

*a*+

*ih*),

*i*= 0, 1, 2,…, 2

*n*.

The derivation of Simpson’s rule is based on the replacement of the integrand *f*(*x*) on each of the closed intervals *[a* + 2*hk*, *a* + 2*h*(*k* + 1)], *k* = 0, 1,…, *n* - 1, by a corresponding interpolation polynomial of the second degree (*see*INTERPOLATION FORMULAS). In geometric terms, the curve described by the equation *y* = *f*(*x*) is replaced by an approximating curve consisting of segments of parabolas. The error resulting from the use of Simpson’s rule is

where *a* ≤ *ξ* ≤ *b*. If *f*(*x*) is a polynomial of degree *m* ≤ 3, then Simpson’s rule is not approximate but exact, since in this case *f*^{(4)}(x) = 0.

Simpson’s rule is named after T. Simpson, who devised it in 1743. The rule was, however, known earlier. For example, it was given by J. Gregory in 1668.

Other formulas for the approximate calculation of definite integrals are discussed in the article APPROXIMATE INTEGRATION.