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  "body": "\n## Summary\n\nGauss–Legendre quadrature is [[Gaussian quadrature|FP37BTP93K]] for the constant weight $1$ on the interval $[-1, 1]$.\nIts $n$ nodes are the zeros of the [[Legendre polynomial|QFYGJVA996]] $P_n$, and it integrates every polynomial of degree at most $2n - 1$ exactly.\nAn integral over any bounded interval $[a, b]$ is handled by a linear change of variable.\nIt is the default Gaussian rule for smooth functions on a bounded interval.\n\n## Explanations\n\n### L1 human\n\nTo integrate a function over $[-1, 1]$, sample it at a few special points and add the samples with special weights.\nThe points are not equally spaced: they crowd towards the two ends of the interval.\nWith two points, at $\\pm 1/\\sqrt{3}$, and weights $1$, the rule already gives the exact integral of every cubic.\nFor example, for $f(x) = x^2$ it gives $1/3 + 1/3 = 2/3$, which is $\\int_{-1}^{1} x^2\\,dx$.\n",
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