Many modern calculus books also include Simpson's Rule. With the notation of the previous section and \(n = 2,\) Simpson's Rule says: \[\int^b_a f(x) \, dx \approx \frac{\omega}{3}[f(x_0) + 4f(x_1) + f(x_2)].\] This rule can be derived by fitting a parabola to the three points \((x_0,f(x_0))\), \((x_1,f(x_1))\) and \((x_2,f(x_2))\), and using the area under that parabola between \(x=a\) and \(x=b\) to approximate the area under \(y=f(x)\). The Trapezoidal Rule and Simpson's Rule are just the first two cases (\(n=1\) and 2, respectively) of a family of formulas known as the Closed Newton-Cotes Formulas. In [Kramp 1815b] and [Bérard 1816], the authors derive the Newton-Cotes formulas for many values of \(n\), and give different answers in the case of \(n=12\).
There is a unique polynomial of degree \(\leq n\) that passes through the \(n+1\) distinct points \[ (x_0,f(x_0)), \; (x_1,f(x_1)), \; \ldots, \; (x_n,f(x_n)).\] The Newton-Cotes formulas are derived by finding the polynomial in a useful form and integrating it to approximate the integral of \(f(x)\). The existence of such a polynomial was proven by Lagrange in 1795 and the form in which he gave it is called the Lagrange Interpolating Polynomial [Lagrange 1795, p. 276ff]. For the \(n+1\) given points, let \[L_{n,k} = \frac{\prod^n_{i=0, i\ne k}(x-x_i)}{\prod^n_{i=0, i\ne k}(x_k-x_i)},\] for \(k=0, 1,\ldots n\). Notice that \(L_{n,k}(x_j) = \delta_{j,k}\) in terms of Kronecker's delta function. Therefore \[P(x) = \sum_{k=0}^n f(x_k)L_{n,k}(x) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{(II)}\] satisfies \(P(x_k) = f(x_k)\) for \(k=0, 1,\ldots n\).
The achievements of Kramp and Bérard in deriving Newton-Cotes formulas for large \(n\) may be impressive, but as a matter of practice, such formulas are almost never used. Not only are they inconvenient to use, but the polynomial (II) has an oscillatory nature that limits the accuracy of these formulas for large \(n\). Instead, modern numerical integration is often based on the composite version of Simpson's Rule inside an adaptive algorithm, such as Romberg Integration; see e.g. [Burden 2016, p. 207].
It's not clear if Kramp and Bérard knew of the work of Roger Cotes (1682–1716). He was an English mathematician, a fellow of Trinity College, Cambridge and the editor of the second edition of Newton's Principia. He published only one paper in his lifetime, but after his untimely death at the age of 33, his papers were published posthumously [Cotes 1722]. In the last part of this book (Opera Miscellanea), Cotes described how to derive the Newton-Cotes formulas, and on page 33 gave the specific formulas for \(n=3,4, \ldots 11\).