
\subsection{\bf Use of Bounding Boxes}
\subsection{\bf to Calculate $\pi$ --- II}

\begin{slide}
\begin{itemize}

\item And calculate
$$
I = \int^1_{-1} dx \int^1_{-1} dy\, g'(x,y)
$$
which is an elementary Monte Carlo integral with $x$ and $y$
generated independently between $-1$ and 1.

\item Actually, the brute force method (generate $x$ uniformly
between $-1$ and 1, $y$ uniformly between $\pm\sqrt{1-x^2}$) is
more efficient---in the language of (NI.16) this second method
has a smaller variance.

\item However, use of bounding boxes is an important technique for it
is often impossible to find ranges analytically while it is easy to
decide if a given $(x,y)$ lies within range of integration.

\centerline{\psfig{figure=ni.fig9.ps,clip=true,width=2.0in}}

\end{itemize}
\end{slide}

