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 degree 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 $P=\left(a=x_0,x_1,...,x_N=b\right)$ of the interval $\left(a\,b\right)$, the $N$th degree Newton-Cotes formula approximates the integral on each subinterval $\left(x_{i-1}\,x_i\right)$ by integrating the $N$th degree polynomial which interpolates $N-1$ 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 degree.

By default, the interval is divided into $10$ equal-sized subintervals.

For the options opts, see the ApproximateInt help page.

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

$\mathrm{int}\left(\sin \left(x\right)\,x=0...5.0\right)$

$0.7163378145$

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right)\:$

$\mathrm{ApproximateInt}\left(\sin \left(x\right)\,x=0...5.0\,\mathrm{method}=\mathrm{newtoncotes}\left[1\right]\right)$

$0.7013515555$

$\mathrm{ApproximateInt}\left(\sin \left(x\right)\,x=0...5.0\,\mathrm{method}=\mathrm{newtoncotes}\left[2\right]\right)$

$0.7163534765$

$\mathrm{ApproximateInt}\left(\sin \left(x\right)\,x=0...5.0\,\mathrm{method}=\mathrm{newtoncotes}\left[3\right]\right)$

$0.7163447696$

$\mathrm{ApproximateInt}\left(\sin \left(x\right)\,x=0...5.0\,\mathrm{method}=\mathrm{newtoncotes}\left[4\right]\right)$

$0.7163378087$

$\mathrm{ApproximateInt}\left(\sin \left(x\right)\,x=0...5.0\,\mathrm{method}=\mathrm{newtoncotes}\left[6\right]\right)$

$0.7163378145$

$\mathrm{ApproximateInt}\left(x\left(x-2\right)\left(x-3\right)\,0..5\,\mathrm{method}=\mathrm{simpson}\,\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{ApproximateInt}\left(\tan \left(x\right)-2x\,x=-1..1\,\mathrm{method}=\mathrm{simpson}\,\mathrm{output}=\mathrm{plot}\,\mathrm{partition}=50\right)$

$\mathrm{ApproximateInt}\left(\ln \left(x\right)\,x=1..100\,\mathrm{method}=\mathrm{simpson}\,\mathrm{output}=\mathrm{animation}\right)$