Chapter 8
Integral Calculus and Its Uses
8.3 Numerical Approximation of Integrals
8.3.3 Simpson's Rule
The patterns in Table 3 suggest another good idea. As we observed, it appears that MR and TR tend to estimate on opposite sides of the exact answer and that the midpoint rule is roughly twice as good as the trapezoidal rule for a given \(n\). That is, the error in MR is approximately half that of TR. These observations suggest that we can expect to find the value of the integral between MR and TR, about twice as far from TR as from MR - in other words, one-third of the way from MR to TR. What we want is an average that weights MR twice as much as TR:
\(\displaystyle\frac{2}{3}\) MR \(\displaystyle + \frac{1}{3}\) TR.
This good idea also has a name: Simpson's Rule, abbreviated SR. How good is Simpson's Rule? In Table 4 we display the results that correspond to those in the two preceding tables - same function, same interval, and same values of \(n\).
| Table 4 Approximations to \(\displaystyle\int_1^{\,3} \frac{1}{x} dx = \ln(3) \approx 1.098612288668110\) |
Evidently Simpson's Rule is such a good idea that even a relatively short sum can approximate a "reasonable" integral to six-place accuracy, and somewhat longer sums - well within the range of computers or calculators - can do much better. Indeed, each increase in the number of steps by a factor of \(10\) decreases the error by a factor of \(1/10,000\)! It is very likely that your calculator carries out Simpson's Rule or something very much like it. And the same is likely to be true of commercial computer software that evaluates integrals.
