\subsection{\bf Change of Variable Before}
\subsection{\bf Numerical Integration---I}

\begin{slide}

\begin{itemize}
\item {\bf Strategy~III}

\begin{description}
\item{$\bullet$} We can also use change of variables in $\int f(x)$ with
$f(x) = {1\over \sqrt{x}} \exp \left( -x^3 \right)$.  

\item{$\bullet$} Namely, if
\begin{eqnarray*}
{1\over\sqrt{x}}\,dx & = & dy \\
y & = & 2\sqrt{x} \\
{1\over\sqrt{x}}\exp\left( -x^3 \right)\,dx & = & \int\exp\left(
- \left( {y\over 2}\right) ^6\right)\,dy \\
\end{eqnarray*}

\item{$\bullet$} which can be done reliably numerically by
approximating integrand by a polynomial, as in standard formulae.

\end{description}
\end{itemize}
\end{slide}


