1 Numerical integration

The aim in this Section is to describe numerical methods for approximating integrals of the form

${\int \; <!--nolimits\; -->}_a^{b}f\left(x\right)dx$

One motivation for this is in the material on probability that appears in HELM booklet  39. Normal distributions can be analysed by working out

${\int \; <!--nolimits\; -->}_a^b\frac{1}{\sqrt{2\pi }}{e}^{-x^2∕2}dx$

for certain values of $a$ and $b$ . It turns out that it is not possible, using the kinds of functions most engineers would care to know about, to write down a function with derivative equal to $\frac{1}{\sqrt{2\pi }}{e}^{-x^2∕2}$ so values of the integral are approximated instead. Tables of numbers giving the value of this integral for different interval widths appeared at the end of HELM booklet  39, and it is known that these tables are accurate to the number of decimal places given. How can this be known? One aim of this Section is to give a possible answer to that question.

It is clear that, not only do we need a way of approximating integrals, but we also need a way of working out the accuracy of the approximations if we are to be sure that our tables of numbers are to be relied on.

In this Section we address both of these points, begining with a simple approximation method.