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  "body": "\n## Definition\n\nLet $a \u003c b$ and let $n \\ge 1$ be an integer.\nThe *closed Newton–Cotes formula* with $n + 1$ nodes on $[a, b]$ is the [[quadrature rule|ADSVMCY77A]]\n$$Q_n(f) = \\sum_{i=0}^{n} w_i\\, f(x_i), \\qquad x_i = a + i h, \\quad h = \\frac{b - a}{n}, \\qquad w_i = \\int_a^b \\ell_i(x)\\, dx,$$\nwhere $\\ell_0, \\dots, \\ell_n$ are the [[Lagrange basis polynomials|DA3CQJ5MJ6]] for the nodes $x_0, \\dots, x_n$.\nFor $n \\ge 0$, the *open Newton–Cotes formula* with $n + 1$ nodes uses instead the nodes $x_i = a + i h$, $i = 1, \\dots, n + 1$, with $h = (b - a)/(n + 2)$, and the weights $\\int_a^b \\ell_i(x)\\, dx$ for these nodes.\n\n## Remarks\n\nSo $Q_n(f)$ is the integral of the interpolating polynomial of $f$ at the nodes, and it is exact for polynomials of degree at most $n$.\nThe closed formula has [[degree of exactness|ZYT1JNT1Y0]] $n$ for odd $n$ and $n + 1$ for even $n$, by symmetry of the nodes.\nExample: $n = 1$ 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.\nNon-example: the two-point Gauss–Legendre rule on $[-1, 1]$ has nodes $\\pm 1/\\sqrt{3}$ and weights $1$.\nIt is interpolatory but not a Newton–Cotes formula.\nIts nodes are neither the endpoints $\\pm 1$, as a closed formula needs, nor $\\pm 1/3$, as the open formula with two nodes needs.\nFor 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.\n",
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