Section Summary

In this section we have studied what your computer (or your calculator) might be doing when it calculates definite integrals numerically. If \(f\) is a continuous function on the interval \([a,b]\), \(n\) is a positive integer, \(\Delta x = \frac{b-a}{n}\), and \(x_k = a+k \Delta x\) for \(k=1, 2, ..., n\), then the definition of the definite integral implies that the left-hand sum (LHS) and the right-hand sum (RHS) approximate the value of the integral:

and

However, these sums require too many calculations for reasonable accuracy in reasonable time. In fact, to get one more decimal place of accuracy with either LHS or RHS, \(n\) has to be increased by a factor of \(10\).

The next step in improving both accuracy and efficiency of calculation is either to average the function values at the endpoints of the subintervals - the Trapezoidal Rule (TR) - or to evaluate \(f\) at the midpoint of each subinterval - the Midpoint Rule (MR):

and

For each of these methods, increasing \(n\) by a factor of \(10\) tends to decrease error by a factor of \(1/10^2\).

Simpson's Rule (SR) is a weighted average of the Midpoint and Trapezoidal Rules:

SR \(\displaystyle=\frac{2}{3}\) MR \(\displaystyle+ \frac{1}{3}\) TR.

For SR, increasing \(n\) by a factor of \(10\) tends to decrease error by a factor of \(1/10^4\). This makes Simpson's Rule an effective tool for rapid calculation of highly accurate approximations to definite integrals. The program in your computer or calculator is likely to be an adaptation of Simpson's Rule or something very much like it.