Solution

Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 2 / 18D / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 2 / 2I / ii / Solution

{title2=$A_j=\int\ell_j^2>0$}

For an $n$-node <Gaussian quadrature> rule exact through degree $2n-1$, the cardinal <Lagrange interpolation polynomial> $\ell_j$ has square of degree $2n-2$. Exactness gives $A_j=\int\ell_j^2>0$ for a positive integration weight. This proof turns <polynomial> exactness directly into positivity.

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 Incoming links: Positive weights of Gaussian quadrature