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  "body": "\n## Statement\n\nLet $h \u003e 0$, let $x_k = x_0 + kh$ for $k = 0, \\ldots, 4$, and let $f \\in C^6[x_0, x_4]$ with $f_k = f(x_k)$.\nThen there is $\\xi \\in (x_0, x_4)$ with\n$$\\int_{x_0}^{x_4} f(x) \\, dx = \\frac{2h}{45} \\big( 7 f_0 + 32 f_1 + 12 f_2 + 32 f_3 + 7 f_4 \\big) - \\frac{8 h^7}{945} f^{(6)}(\\xi).$$\n\n## Remarks\n\nThe rule is the closed [[Newton–Cotes formula|BQPQAY1FDJ]] with five nodes: its weights are the integrals of the Lagrange basis polynomials of the five equally spaced nodes.\nBy symmetry it is exact for all polynomials of degree at most $5$, one more than the interpolation degree, so its [[degree of exactness|ZYT1JNT1Y0]] is $5$.\nThe error constant can be read off from $f(x) = x^6$ on $[0, 4]$ with $h = 1$: the integral is $\\frac{4^7}{7}$, the rule gives $\\frac{4^7}{7} + \\frac{128}{21}$, and $\\frac{8}{945} \\cdot 720 = \\frac{128}{21}$.\nCheck of the weights: $\\frac{2h}{45}(7 + 32 + 12 + 32 + 7) = 4h$, the length of the interval.\nThe composite rule on $4m$ subintervals has error $O(h^6)$ for $f \\in C^6$.\n\n## Explanations\n\n### L1 human\n\nBoole's rule estimates an integral from five equally spaced samples.\nIt fits a polynomial of degree four through the samples and integrates that polynomial instead.\nThe weights $7, 32, 12, 32, 7$ come out of this fit, and the rule is exact for all polynomials up to degree five.\n\n### L2 human\n\nBoole's rule is the next member after the trapezoidal rule, Simpson's rule and Simpson's $3/8$ rule in the closed Newton–Cotes family.\nIt equals one step of [[Richardson extrapolation|WMTZG71K2Y]] applied to Simpson's rule with steps $2h$ and $h$, as in the Romberg method.\nHigher Newton–Cotes rules eventually have negative weights and become unstable, so in practice one uses composite rules of low order or Gaussian rules instead.\n",
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