Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / A / 2 / c / Solution

For weight $x^2$, the moments are ${\int }_0^1{x}^{j+2}dx=1/(j+3)$. Monic orthogonalization gives ${\varphi }_0=1$ and ${\varphi }_1=x-(1/4)/(1/3)=x-3/4$.

For ${\varphi }_2=x^2+ax+b$, orthogonality to $1,x$ requiresSolving gives $a=-4/3$, $b=2/5$. HenceThese are the nodes for the two-node Gaussian quadrature with quadratic weight.