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Application to the case N=2

If we wish to integrate \begin{equation*} \int_{-1}^1f(x)dx, \end{equation*} with f(x)=x^2 , we approximate \begin{equation*} I=\int_{-1}^1x^2dx \approx \sum_{i=0}^{N-1}\omega_ix_i^2. \end{equation*}