Appendix
A comparison of area estimates, and Simpson's rule
We have mentioned several different ways of estimating the area under a curve: left-endpoint, right-endpoint, midpoint and trapezoidal estimates. It's interesting to compare them.
Recall that we consider the function \(f(x)\) over the interval \([a,b]\) and divide the interval into \(n\) subintervals \([x_0, x_1], [x_1, x_2], \dots, [x_{n-1}, x_n]\) each of width \(\Delta x = \frac{1}{n} (b-a)\).
All of the estimates discussed give the approximate area under \(y=f(x)\) in the form of a sum of products of values of \(f\) with the width \(\Delta x\). In fact, the left-endpoint, right-endpoint and trapezoidal estimates are all of the form
\[ \big[ c_0 f(x_0) + c_1 f(x_1) + c_2 f(x_2) + \dots + c_{n-1} f(x_{n-1}) + c_n f(x_n) \big] \, \Delta x. \](The midpoint rule is slightly different, since it evaluates the function \(f\) at the midpoints of subintervals.) The coefficients \(c_j\) are compared in the following table.
| Estimate | \(c_0\) | \(c_1\) | \(c_2\) | \(c_3\) | \(c_4\) | \(\dots\) | \(c_{n-2}\) | \(c_{n-1}\) | \(c_n\) |
|---|---|---|---|---|---|---|---|---|---|
| left endpoint | 1 | 1 | 1 | 1 | 1 | \(\dots\) | 1 | 1 | 0 |
| right endpoint | 0 | 1 | 1 | 1 | 1 | \(\dots\) | 1 | 1 | 1 |
| trapezoidal | \(\frac{1}{2}\) | 1 | 1 | 1 | 1 | \(\dots\) | 1 | 1 | \(\frac{1}{2}\) |
| Simpson | \(\frac{1}{3}\) | \(\frac{4}{3}\) | \(\frac{2}{3}\) | \(\frac{4}{3}\) | \(\frac{2}{3}\) | \(\dots\) | \(\frac{2}{3}\) | \(\frac{4}{3}\) | \(\frac{1}{3}\) |
There is another estimate, called Simpson's rule, which takes a slightly different set of coefficients, as shown in the table. It requires that the number of subintervals \(n\) is even. The coefficients start at \(\frac{1}{3}\), then alternate between \(\frac{4}{3}\) and \(\frac{2}{3}\), before ending again at \(\frac{1}{3}\). (It is as if every second coefficient in the list for the trapezoidal estimate donated \(\frac{1}{6}\) to its~neighbours.)
Explicitly, the estimate from Simpson's rule for the area under \(y=f(x)\) between \(x=a\) and \(x=b\) is
\[ \Big[ \frac{1}{3} f(x_0) + \frac{4}{3} f(x_1) + \frac{2}{3} f(x_2) + \frac{4}{3} f(x_3) + \frac{2}{3} f(x_4) + \dots + \frac{2}{3} f(x_{n-2}) + \frac{4}{3} f(x_{n-1}) + \frac{1}{3} f(x_n) \Big] \Delta x. \]Although the coefficients may appear somewhat bizarre, Simpson's rule almost always gives a much more accurate answer than any of the other estimates mentioned. We'll see why in the next section.
Next page - Appendix - Exactness of area estimates
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