Solution
= Solution
Suppose that an $(n+1)$-node <quadrature rule> were exact on $\mathcal P_{2n+2}$. The nodal polynomial $Q_{n+1}$ has degree $n+1$, so $Q_{n+1}^2\in\mathcal P_{2n+2}$. At every node it vanishes, and hence
$$
I_n(Q_{n+1}^2)=0.
$$
But $Q_{n+1}^2$ is a nonzero nonnegative <polynomial>, and the weight function is positive on $(a,b)$. Therefore
$$
I(Q_{n+1}^2)=\int_a^bQ_{n+1}(x)^2w(x)\,dx>0,
$$
a contradiction. This is the <degree ceiling for quadrature exactness>.