The ApproximateInt(f(x), x = a..b, method = simpson[3/8], opts) command approximates the integral of f(x) from a to b by using Simpson's 3/8 rule.  This rule is also known as Newton's 3/8 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 $P=\left(a=x_0,x_1,...,x_N=b\right)$ of the interval $\left(a\,b\right)$, Simpson's 3/8 rule approximates the integral on each subinterval $\left(x_{i-1}\,x_i\right)$ by integrating the cubic function that interpolates the four points $\left(x_{i-1}\,f\left(x_{i-1}\right)\right)$, $\left(\frac{2x_{i-1}}{3}+\frac{x_i}{3}\,f\left(\frac{2x_{i-1}}{3}+\frac{x_i}{3}\right)\right)$, $\left(\frac{x_{i-1}}{3}+\frac{2x_i}{3}\,f\left(\frac{x_{i-1}}{3}+\frac{2x_i}{3}\right)\right)$, and $\left(x_i\,f\left(x_i\right)\right)$.  This value is

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

Traditionally, Simpson's 3/8 rule is written as: given N, where N is a positive multiple of 3, and given equally spaced points $a=x_0,x_1,x_2,...,x_N=b$, an approximation to the integral ${\int }_a^{b}f\left(x\right)\ⅆx$ is

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.

$\mathrm{polynomial}≔\mathrm{CurveFitting}\left[\mathrm{PolynomialInterpolation}\right]\left(\left[x\left[0\right]\,\frac{2x\left[0\right]+x\left[1\right]}{3}\,\frac{x\left[0\right]+2x\left[1\right]}{3}\,x\left[1\right]\right]\,\left[f\left(0\right)\,f\left(\frac{1}{3}\right)\,f\left(\frac{2}{3}\right)\,f\left(1\right)\right]\,z\right)\:$

$\mathrm{integrated}≔\mathrm{int}\left(\mathrm{polynomial}\,z=x\left[0\right]..x\left[1\right]\right)\:$

$\mathrm{factor}\left(\mathrm{integrated}\right)$

$-\frac{\left(x_0-x_1\right)\left(f\left(0\right)+f\left(1\right)+3f\left(\frac{1}{3}\right)+3f\left(\frac{2}{3}\right)\right)}{8}$

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

$\mathrm{ApproximateInt}\left(\sin \left(x\right)\,x=0..5\,\mathrm{method}=\mathrm{simpson}\left[\frac{3}{8}\right]\right)$

$\frac{3\sin \left(\frac{23}{6}\right)}{16}+\frac{\sin \left(4\right)}{8}+\frac{3\sin \left(\frac{25}{6}\right)}{16}+\frac{3\sin \left(\frac{13}{3}\right)}{16}+\frac{\sin \left(\frac{9}{2}\right)}{8}+\frac{3\sin \left(\frac{14}{3}\right)}{16}+\frac{3\sin \left(\frac{29}{6}\right)}{16}+\frac{\sin \left(5\right)}{16}+\frac{3\sin \left(\frac{8}{3}\right)}{16}+\frac{3\sin \left(\frac{17}{6}\right)}{16}+\frac{\sin \left(3\right)}{8}+\frac{3\sin \left(\frac{19}{6}\right)}{16}+\frac{3\sin \left(\frac{10}{3}\right)}{16}+\frac{\sin \left(\frac{7}{2}\right)}{8}+\frac{3\sin \left(\frac{11}{3}\right)}{16}+\frac{3\sin \left(\frac{5}{3}\right)}{16}+\frac{3\sin \left(\frac{11}{6}\right)}{16}+\frac{\sin \left(2\right)}{8}+\frac{3\sin \left(\frac{13}{6}\right)}{16}+\frac{3\sin \left(\frac{7}{3}\right)}{16}+\frac{\sin \left(\frac{5}{2}\right)}{8}+\frac{3\sin \left(\frac{1}{3}\right)}{16}+\frac{\sin \left(\frac{1}{2}\right)}{8}+\frac{3\sin \left(\frac{2}{3}\right)}{16}+\frac{3\sin \left(\frac{5}{6}\right)}{16}+\frac{3\sin \left(\frac{7}{6}\right)}{16}+\frac{3\sin \left(\frac{4}{3}\right)}{16}+\frac{\sin \left(\frac{3}{2}\right)}{8}+\frac{3\sin \left(\frac{1}{6}\right)}{16}+\frac{\sin \left(1\right)}{8}$

$\mathrm{ApproximateInt}\left(x\left(x-2\right)\left(x-3\right)\,x=0..5\,\mathrm{method}=\mathrm{simpson}\left[\frac{3}{8}\right]\,\mathrm{output}=\mathrm{plot}\right)$

$\mathrm{ApproximateInt}\left(\tan \left(x\right)-2x\,x=-1..1\,\mathrm{method}=\mathrm{simpson}\left[\frac{3}{8}\right]\,\mathrm{output}=\mathrm{plot}\,\mathrm{partition}=50\right)$

$\mathrm{ApproximateInt}\left(\ln \left(x\right)\,1..100\,\mathrm{method}=\mathrm{simpson}\left[\frac{3}{8}\right]\,\mathrm{output}=\mathrm{animation}\right)$