Degree of exactness of a quadrature rule

Remarks

By linearity, clause 1 holds if and only if $Q(x^k) = I(x^k)$ for $k = 0, 1, \dots, d$. The 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 $Q(p) = 0 < I(p)$. A rule has degree of exactness at least $n - 1$ if and only if it is interpolatory. Example: 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 $(b - a)^3/12$. Non-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 $0$. Some authors call it the degree of precision or the algebraic degree of accuracy.