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

\begin{slide}

\begin{itemize}
\item Let us consider implications of the transformation $x\rightarrow
y = 2\sqrt{x}$.

\item Integration formulae for $x$ (which assume integrand $f(x)\sim$
polynomial) $\Rightarrow$ equally spaced (or distribution of Gaussian
formulae) grid points.

\item But we have seen $y$ is a better variable and we will rather get
equally spaced points in $y$.

\item Now $dx = \sqrt{x}\,dy$.  A fixed spacing $dy\Rightarrow$ spacing
in $x$ space $\propto\sqrt{x}$.

\item That is, $x$ points are concentrated near the origin.

\item This is intuitively reasonable as the integrand
${1\over\sqrt{x}}\exp\left( -x^3 \right)$ is rapidly varying near the
origin.
\end{itemize}

\end{slide}

