Interpolatory quadrature ($Q(f)={\int }_a^b{p}_n(x)dx$)

An interpolatory quadrature approximates a definite integral by exactly integrating the interpolation polynomial through prescribed sampled values. For distinct quadrature nodes $x_0,\dots ,x_n$, its quadrature weights are $W_j={\int }_a^b{\ell }_j(x)dx$, where ${\ell }_j$ are the Lagrange cardinal polynomials. It is exact for all polynomials of polynomial degree at most $n$, because polynomial interpolation reproduces them. Newton-Cotes quadrature adds equal spacing of the quadrature nodes.

A Newton-Cotes quadrature is an interpolatory quadrature rule whose quadrature nodes are equally spaced. Integrating the Lagrange interpolation polynomial gives quadrature weights $W_j={\int }_a^b{\ell }_j(x)dx$. A Newton-Cotes closed quadrature includes both endpoints; an open rule uses interior quadrature nodes. Simpson's rule is the closed rule with three quadrature nodes. Its quadrature weights are $(b-a)(1,4,1)/6$.