The ApproximateInt(f(x), x = a..b, method = newtoncotes[N], opts) command approximates the integral of f(x) from a to b by using the Nth order Newton-Cotes formula. 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 , the Nth order Newton-Cotes formula approximates the integral on each subinterval  by integrating the Nth order polynomial which interpolates  equally spaced points between the end points of the interval.

The Newton-Cotes formulae are generalizations of the simpler polynomial interpolation routines.  The following table gives the correspondence between the other methods and the order.

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.