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  "body": "\n## Definition\n\nLet $Q(f) = \\sum_{i=1}^{n} w_i f(x_i)$ be a [[quadrature rule|ADSVMCY77A]] on $[a, b]$, and let $I(f) = \\int_a^b f(x)\\, dx$.\nThe rule has *degree of exactness* $d \\ge 0$ if:\n\n1. $Q(p) = I(p)$ for every real [[polynomial|MAM9H3SY3Y]] $p$ of [[degree|0DPF6WRZW6]] at most $d$;\n2. $Q(p) \\ne I(p)$ for some polynomial $p$ of degree $d + 1$.\n\n## Remarks\n\nBy linearity, clause 1 holds if and only if $Q(x^k) = I(x^k)$ for $k = 0, 1, \\dots, d$.\nThe degree of exactness of an $n$-node rule is at most $2n - 1$: for $p(x) = \\prod_i (x - x_i)^2$, of degree $2n$, one has $Q(p) = 0 \u003c I(p)$.\nA rule has degree of exactness at least $n - 1$ if and only if it is interpolatory.\nExample: the midpoint rule $Q(f) = (b - a) f(m)$ with $m = (a + b)/2$ has degree $1$: it is exact for $1$ and $x - m$, while for $(x - m)^2$ it gives $0$ instead of $(b - a)^3/12$.\nNon-example: the left endpoint rule $Q(f) = (b - a) f(a)$ does not have degree $1$, since for $p(x) = x - a$ it gives $0$ instead of $(b - a)^2/2$; its degree is $0$.\nSome authors call it the degree of precision or the algebraic degree of accuracy.\n",
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