Simpson's rule is a repetitive process for estimating the definite integral of a function (the area underneath the curve). Most calculators use a modification of this rule. As mentioned earlier, it is repetitive -- and it must be repeated for an even number of iterations. It is generally more accurate than the othe repetitive approximation of the definite integral -- the Trapezoidal Rule.

What is this wonderful rule? Here it is, using Integ for the integral sign (the formatting might be slighly messed up, due to the difficulty of putting this in pASCII].):

Integ(a, b, f(x) dx) ~=
b-a [f(x0)+4f(x1)+2(fx2)+4(fx3)...+4f(xn-a)+f(xn)]
3n

The idea behind all this is that we use portions of the graph in the form y=cx^2+dx+e for constants c d and e.

The error from Simposon's rule can be found with the following formula:

M(b-a)^5
180n^4

M here is a positive real number such that |f(4)(x)| <= M for every x in [a,b].

Simpson's rule is still very useful for finding definite integrals using calculators. Many programs that will compute a definite integral using Simpson's rule are available from ticalc.org or hpcalc.org.