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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $F$ be a class of [[integrable functions|RSAK8YHMXF]] $f \\colon [a, b] \\to \\R$, with the [[definite integral|021WEM680W]] $I(f) = \\int_a^b f(x)\\, dx$.\nA *numerical integration method* on $F$ is a rule that assigns to each $f \\in F$ a number $Q(f)$.\nThe number $Q(f)$ is computed from finitely many values $f(x_1), \\dots, f(x_n)$ at points $x_i \\in [a, b]$, and possibly from finitely many values of derivatives of $f$.\nIts *error* at $f$ is $E(f) = I(f) - Q(f)$.\n*Numerical integration* is the design and analysis of such methods, so that $\\abs{E(f)}$ is small for the functions of interest.\n\n## Remarks\n\nThe same definition applies to integrals over regions of $\\R^d$ or with a weight function; in several dimensions one often speaks of cubature.\nExample: the midpoint method $Q(f) = (b - a) f\\big(\\tfrac{a + b}{2}\\big)$ on continuous functions uses one value; for $f(x) = x^2$ on $[0, 1]$ it gives $1/4$, with error $1/3 - 1/4 = 1/12$.\nNon-example: $Q(f) = F(b) - F(a)$ with an antiderivative $F$ is exact, but it is not a numerical integration method in this sense, because it needs an antiderivative, not finitely many values of $f$.\nThe points $x_i$ may also be chosen adaptively, depending on earlier values, as in adaptive quadrature, or at random, as in Monte Carlo integration.\n",
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