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  "body": "\n## Summary\n\nSimpson's rule approximates $\\int_a^b f$ by integrating the parabola through the values of $f$ at $a$, at the midpoint, and at $b$, which gives $\\frac{b - a}{6}\\big(f(a) + 4 f(m) + f(b)\\big)$.\nIt is the closed [[Newton–Cotes formula|BQPQAY1FDJ]] with three nodes, and it is exact for all cubic polynomials.\nIts composite form has error of order $h^4$ for smooth integrands, which makes it a standard simple method.\n\n## Explanations\n\n### L1 human\n\nInstead of joining samples by straight lines, Simpson's rule fits a parabola through three equally spaced samples and takes the area under it.\nThe middle sample gets weight four times that of the ends, because the parabola's area depends most on the height in the middle.\nFor $f(x) = x^3$ on $[0, 1]$ it gives $\\frac{1}{6}\\big(0 + 4 \\cdot \\frac{1}{8} + 1\\big) = \\frac{1}{4}$, exactly the true value.\nHalving the step size reduces the error by a factor of about sixteen.\n",
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