

\subsection{\bf Use of Bounding Boxes}
\subsection{\bf in Complicated Geometries --- I}

\begin{slide}

\begin{itemize}
\item Example:  Integrate $g(x,y)$ over region $h(x,y)\ge 0$.
$$
h(x,y) = x^3 + 3xy^2 + 23y^3
$$
or something complicated like this.  

\item It is indeed $\underline{\rm easy}$ to decide if $h(x,y) \ge 0$
or $< 0$ for a given value $(x,y)$.

\item However, it is hard to solve equation $x^3 + 3xy^2 + 23y^3 = 0$
(the boundary of domain).  

\item Note you cannot use Simpson/Gauss/Trapezoidal to integrate
\[
\int^{+1}_{-1} \int^{+1}_{-1} dx\,dy\,F(x,y)
\]
$$\vbox{\halign{#\hfil\tabskip=0.5em&#\hfil\tabskip=3em&#\hfil\tabskip=0pt\cr
where\ \ \ \ \ \ \ \ $F(x,y)$ & $= f(x,y)$ & $x^2 + y^2 \le 1$ \cr
&$ = 0$ & $x^2 + y^2 > 1$ \cr
}}$$
\end{itemize}

\centerline{\psfig{figure=ni.fig10.ps,clip=true,width=5.0in}}

\end{slide}

