Servois' 1817 "Memoir on Quadratures" – The Trapezoidal Rule

Most students learn in calculus that a simple approximation to the definite integral is

\[\int^b_a f(x) \, dx \approx \frac{b-a}{2}[f(a) + f(b)].\]

This is called the Trapezoidal Rule, because the right-hand side is the area of the trapezoid bounded by the interval \([a,b]\) on the \(x\)-axis, the lines \(x=a\) and \(x=b\) and the straight line segment joining \((a,f(a))\) to \((b,f(b))\). Although this approximation generally isn't very accurate, it can be greatly improved by breaking the interval \([a,b]\) into subintervals.

In particular, we let \(n\) be a natural number, \(\omega = \frac{b-a}{n}\), and \(x_k = a +k \omega\), for \(k=0, 1, 2, \ldots, n\). Then if we apply the Trapezoid Rule to each subinterval \([x_{k-1},x_k]\), for \(k= 1, 2, \ldots, n\), and sum the results, we have¹

\[\int^b_a f(x) \, dx \approx \frac{\omega}{2}\left[f(x_0) +2 \sum^{n-1}_{k=1}f(x_k) + f(x_n)\right]. \,\,\,\,\,\,\,\,\,\,\,\,\,\,(I)\]

This is called the Composite Trapezoidal Rule, or just the Trapezoidal Rule in some books. Intuitively, the approximation should get better and better with larger values of \(n\). This is indeed the case for nice functions \(f\). For example, under the hypothesis that \(f''(x)\) is bounded on \([a,b]\), the error in the approximation is roughly proportional to \(n^{-2}\); see [Burden 2016, p. 205].

In [Kramp 1815a], the Composite Trapezoidal Rule is applied repeatedly for \(n=1\), 2, 3, 4, 6, and 12, and these six results are substituted into an extrapolating formula that gives a new approximation that is generally much better even than the approximation corresponding to \(n=12\). Kramp showed that this gave surprisingly good estimates (0.69314806 and 0.78539271 respectively) of

\[\ln 2 = \int^2_1 \frac{dx}{x} \quad \mbox{and}\quad \frac{\pi}{4} = \int^1_0 \frac{dx}{1+x^2}.\]

Gergonne criticized Kramp's result both because of the arbitrariness of the extrapolation rule and the fact that, without an error bound, we have no way of knowing how good the approximation is when the true value of the integral is unknown–and if the value is known, then there's no need for an approximation procedure!

Notes:

1. Equation numbers in this article are given in Roman numerals to distinguish them from the original equation numbers used by Servois in [Servois 1817]. There is no correlation between these Roman numerals and his Hindu Arabic numerals.