The ApproximateInt(f(x), x = a..b, method = simpson, opts) command approximates the integral of f(x) from a to b by using Simpson's rule. The first two arguments (function expression and range) can be replaced by a definite integral.

If the independent variable can be uniquely determined from the expression, the parameter x need not be included in the calling sequence.

Given a partition  of the interval , Simpson's rule approximates the integral on each subinterval  by integrating the quadratic function that interpolates the three points , , and .  This value is

In the case that the widths of the subintervals are equal, the approximation can be written as

Traditionally, Simpson's rule is written as: given N where N is an even integer and given equally spaced points , an approximation to the integral  is

By default, the interval is divided into  equal-sized subintervals.

For the options opts, see the ApproximateInt help page.

This rule can be applied interactively, through the ApproximateInt Tutor.