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

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 $(b-a)(1,4,1)/6$.