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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.

Integrate the Lagrange interpolation polynomial and exchange the integral with the finite sum:Therefore the interpolatory quadrature weights areThe resulting quadrature rule is exact for every polynomial of degree at most $n$, since its interpolant is itself. For a general continuous $f$, replacing $f$ by its interpolant supplies the approximation.