Simpson's rule is a Newton-Cotes formula for approximating the integral of a function  using quadratic polynomials (i.e., parabolic arcs instead of the straight line segments used in the trapezoidal rule). Simpson's rule can be derived by integrating a third-order Lagrange interpolating polynomial fit to the function at three equally spaced points. In particular, let the function  be tabulated at points , , and  equally spaced by distance , and denote . Then Simpson's rule states that

			(1)
			(2)

Since it uses quadratic polynomials to approximate functions, Simpson's rule actually gives exact results when approximating integrals of polynomials up to cubic degree.

For example, consider  (black curve) on the interval , so that , , and . Then Simpson's rule (which corresponds to the area under the blue curve obtained from the third-order interpolating polynomial) gives

			(3)
			(4)
			(5)

whereas the trapezoidal rule (area under the red curve) gives  and the actual answer is 1.

In exact form,

			(6)
			(7)

where the remainder term can be written as

(8)

with  being some value of  in the interval .

An extended version of the rule can be written for  tabulated at , , ...,  as

(9)

where the remainder term is

(10)

for some .

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 886, 1972.

Horwitz, A. "A Version of Simpson's Rule for Multiple Integrals." J. Comput. Appl. Math. 134, 1-11, 2001.

Jeffreys, H. and Jeffreys, B. S. Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 286, 1988.

Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, p. 105, 2004. http://www.mathematicaguidebooks.org/.

Whittaker, E. T. and Robinson, G. "The Trapezoidal and Parabolic Rules." The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 156-158, 1967.