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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*}