Newton–Cotes formulas

Remarks

So $Q_n(f)$ is the integral of the interpolating polynomial of $f$ at the nodes, and it is exact for polynomials of degree at most $n$. The closed formula has degree of exactness $n$ for odd $n$ and $n + 1$ for even $n$, by symmetry of the nodes. Example: $n = 1$ gives the trapezoidal rule $\frac{h}{2}(f(a) + f(b))$, $n = 2$ gives Simpson's rule $\frac{h}{3}(f_0 + 4 f_1 + f_2)$, and $n = 3$ gives Simpson's $3/8$ rule $\frac{3h}{8}(f_0 + 3 f_1 + 3 f_2 + f_3)$; the open formula with $n = 0$ is the midpoint rule. Non-example: the two-point Gauss–Legendre rule on $[-1, 1]$ has nodes $\pm 1/\sqrt{3}$ and weights $1$. It is interpolatory but not a Newton–Cotes formula. Its nodes are neither the endpoints $\pm 1$, as a closed formula needs, nor $\pm 1/3$, as the open formula with two nodes needs. For closed formulas with $n \ge 8$ some weights are negative, and as $n \to \infty$ the formulas need not converge even for analytic integrands, much as in Runge's phenomenon.