If it is undesirable (for example, when using
tables) to increase the subdivision of an interval 
, in
order to improve the accuracy of a quadrature, one
alternative is to use an approximating polynomial of higher degree. The integration formula, based on a quadratic (i.e., parabolic)
approximation is called Simpson's Rule. It is adequate for most purposes that one is likely to
encounter in practice.
Simpson's Rule gives for 
.
A parabolic arc is fitted to the curve y = f(x) at the
three tabular points Hence, if N
- (b - a) is even, one obtains Simpson's Rule:
,
where
.
Integration by Simpson's Rule involves computation of a finite sum of given values of the integrand f, as in the case of the trapezoidal rule. Simpson's Rule is also effective for implementation on a computer; a single direct application in a hand calculation usually gives sufficient accuracy.
For a given integrand f, it is quite appropriate to computer program increased interval subdivision, in order to achieve a required accuracy, while for hand calculations an error bound may again be useful.
Let the function f(x) have in 
the Taylor
expansion
,
then
.
One may reformulate the quadrature rule for by replacing fj+2
= f (j+1 + h) and fj
= f (xj+1 - k) by its
Taylor
series; thus
.
A comparison of these two versions shows that the truncation error is
.
Ignoring higher order terms, we conclude that the approximate bound on this error while estimating
by Simpson's Rule with N/2 subintervals of width 2h is
.
Note that the error bound is proportional to h4, compared with h2 for the cruder trapezoidal rule. Note that Simpson's rule is exact for cubics!
We shall estimate the value of the integral
,
using Simpson's rule and the data in Exercise 2 of STEP29. If we choose h = 0.15 or h = 0.05, there will be an even number of intervals. Denoting the approximation with strip width h by S(h), we obtain
and
,
respectively. Since f(4)(x) = -15x-7/2/16, an approximate truncation error bound is
,
whence it is 0.000 000 8 for h = 0.15 and 0.000 000 01 for h = 0.05. Note that the truncation error is negligible; within round-off error, the estimate is 0.32148(6).
,
.