degree of precision

The highest polynomial degree a quadrature rule integrates exactly, every time -- not just approximately close.

A quadrature rule has degree of precision $d$ if it integrates every polynomial of degree $\le d$ exactly, and fails to integrate at least one polynomial of degree $d+1$ exactly. Simpson's rule fits a parabola (degree $2$ ) through three points but has degree of precision $3$ , since a cubic's odd-degree term cancels by symmetry across the panel; an $n$ -point Gauss-Legendre rule reaches degree of precision $2n-1$ , the maximum possible for $n$ evaluations.

Examples

The trapezoidal rule fits a line (degree 1) but has degree of precision exactly 1, not higher -- unlike Simpson's rule, it gets no free extra degree from symmetry.