= Newton-Cotes closed quadrature
{c}
{title2=$w_j=\int_0^s L_j(t)\,dt$}
= Closed Newton-Cotes quadrature
{synonym}
Integrate the <Lagrange polynomial> through values at all equally spaced nodes $0,\ldots,s$, including both endpoints. The resulting <quadrature rule> is exact through degree $s$. For even $s$, the node <polynomial> $\prod_{j=0}^s(t-j)$ is odd about $s/2$, so its integral vanishes and exactness extends through degree $s+1$. The weights need not remain positive as the number of nodes increases.
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