
\subsection{\bf NI.3:\ \ Gaussian Integration}

\begin{slide}
\begin{itemize}
\item i) We are evaluating the same formula
$$
I = \int^b_a g(x)\,f(x)\,dx \eqno\hbox{(NI.1)}
$$
in terms of the same ansatz
$$
I = \sum^n_{r=0} w_r\, f(x_r) \eqno\hbox{(NI.2)}
$$

\item Now we choose $\underline{\rm both}$ $w_r$ $\underline{\rm and}$
$x_r$ (a grand total of $2n+2$ parameters) to make (NI.2) exact for
polynomials of degree $\le 2n+1$.  ($2n+2$ equations analogous to
(NI.3)).

\item With Newton-Cotes, one only chooses $n+1$ parameters $w_r$
viewing $x_r$ as fixed.

\item Thus, Newton-Cotes formulae are exact for polynomials of degree
$\le n$.
\end{itemize}
\end{slide}

