%Double Integrals
%If you modify this file, please indicate here and in the footnote
%Math263D

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\begin{document}

\begin{center}
{\Large
Double Integrals
\footnote{Copyright \copyright 2002 Steve Chapin, and Todd Young.
All rights reserved. Please address comments to young@math.ohiou.edu.}}
\end{center}

\begin{enumerate}
\item Enter the following commands:\\
      \verb&   format long& \dotfill Sets the number of digits displayed to 15.\\
      \verb&   f = inline('x*y^2')& \dotfill Defines a function.  \\
      \verb&   f(2, 3)&\\
      \verb&   dblquad(f, 0, 1, 0, 1)&\\
 Calculate this double integral by hand to confirm the answer. To get
 an explanation of the command, you may want
 to enter: \verb& help dblquad&


\item Next try to use \verb&dblquad& to integrate the following functions
        on the same domain as above:\\
      \verb&   f = inline('x^2*y^2')&\\
      \verb&   f = inline('x.^2*y^2')&\\
      \verb&   f = inline('x.^2*y.^2')&\\
Calculate this double integral by hand to determine which of the outputs is correct.
You should conclude the following: for the \verb&dblquad& command to work properly,
\verb&x& must be a vector and \verb&y& must be a scalar.


\item The following is a trick that can be used for double integrals on regions that are
        not square:\\
     \verb&   f = inline('(x.^2*y^2).*(x+y <= 1)')&\\
     \verb&   dblquad(f, 0, 1, 0, 1)&\\
This should calculate the integral of $x^2y^2$ 
on the triangle with corners at $(0, 0)$, $(1, 0)$, and $(0,1)$. 
Calculate this integral by hand to confirm the calculation above.

\item Write a brief report including your hand calculations.
         Use complete sentences and standard mathematical notation.


\end{enumerate}

\vfill
\noindent
\textsf{The goal of this project is to familiarize the user
     with the double integration capabilities of
     the program. Just as hand calculations of multiple integrals are tricky,
     so are computer calculations.}

\end{document}
