%Approximate Double Integrals
%Math263D
\documentclass[12pt]{article}
\usepackage{times}
\pagestyle{empty}
\addtolength{\textwidth}{1.2in}
\addtolength{\textheight}{1.2in}
\addtolength{\oddsidemargin}{-.58in}
\addtolength{\evensidemargin}{-.58in}
\renewcommand{\baselinestretch}{1.0}
\parindent = 0cm
\parskip = .1cm
\begin{document}
\begin{center}
{\Large
Approximate Double Integrals\footnote{
Copyright \copyright 2002 Todd Young.
All rights reserved. Please address comments to young@math.ohiou.edu.}}
\end{center}
\begin{enumerate}
\item Go to the O.U. \textsc{MatLab} website (\verb&www.math.ohiou.edu/~matlab&).
In 263D, under this assignment, click on: \verb&lowerleft.m&\\
This will download a program file; save it to your working directory.
\item Open \textsc{MatLab}. In the command window, check the current directory, and
if needed, change to the directory where you saved \verb&lowerleft.m&.
\item In the command window, type the following commands:\\
\verb& format long&\\
\verb& f = inline('x*y^2', 'x', 'y')&\\
\verb& lowerleft(f, 0, 1, 0, 2, 10, 20)&\\
\textsc{MatLab} should return the answer \verb&1.1115000000000&. Calculate the integral
exactly by hand and find the \% error.
\item Click on the file icon at the upper left corner and open the file
\verb&lowerleft.m&, this will be the program that was used in the
command above by the same name. Read the program and comments.
\item Save the program as \verb¢erpoint.m&. Modify it to do Riemann
sums using the centerpoint rather than the lowerleft point. You will
need to change the name of the function in the first line to \verb¢erpoint&,
otherwise, you will need to change very little.
\item Test your new program on $f(x, y) = xy^2$ by typing:\\
\verb& centerpoint(f, 0, 1, 0, 2, 10, 20)&\\
This answer should be closer to the correct answer than \verb&lowerleft&.
Test this using the \% error.
\item Try out your new program on the following integral:
$$
\int_0^2\int_0^5\sqrt{xy + y^5 + x^2} dydx
$$
Also, try this integral using the command \verb& dblquad & as in the previous
homework. For \verb&m& and \verb&n& fairly large, does your program come close to
\textsc{MatLab}'s built in program?
\item Using complete sentences and standard mathematical notation, write a brief report.
\end{enumerate}
\vfill
\noindent
\textsf{This assignment gives students a chance to work with Riemann sums and
gives a very gentle introduction to programming in \textsc{MatLab}.}
\end{document}
