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\Large
Riemann Sums I
\footnote{Copyright \copyright 2005 Todd Young.
All rights reserved. Please address comments to young@math.ohiou.edu.}
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\begin{enumerate}
\item In the command window type:\\
          \verb& dx = .1&\\
          \verb& x = 0:dx:2&\\
          \verb& y = 3*x.^2&\\
      Describe \verb&dx&, \verb&x& and \verb&y&.

\item Next type:\\
          \verb& format long&\\  
          \verb& yl = y(1:20)&\\
          \verb& L = dx*sum(yl)&\\
          \verb& yr = y(2:21)&\\
          \verb& R = dx*sum(yr)&\\
What are \verb&yl& and \verb&yr&? 
What are \verb&L& and \verb&R&? How do they compare with the
true value of the integral (what
is the percentage errors)?

\item Repeat the above commands, but begin with:\\
          \verb& dx = .001&\\
      You will need to adjust the index numbers in the commands
      for \verb&yl& and \verb&yr&.
      What are the percentage errors of these approximations.


\item To obtain a midpoint Riemann sum type:\\
      \verb& dx = .1&\\
      \verb& x = dx/2:dx:2-dx/2&\\
      \verb& y = 3*x.^2&\\
      \verb& M = dx*sum(y)&\\
Explain why this is a midpoint sum. For $dx=.1$ and $dx =.001$
compute the percentage errors. Make a table of 
percentage errors in all the above calculations.



\item Repeat all the above commands for the function 
$f(x) = \sqrt{1 + .5\sin^2{x}}$\\ 
(input as \verb& y = sqrt(1 + .5*sin(x).^2))& and record
the answers. The value of this integral correct to 
15 significant digits is: 2.27220510258726. Make a 
table of percentage errors as above.

\item Using complete sentences and standard mathematical notation, 
      prepare a brief (\verb$< $1 page) written report answering all
      the questions. Do {\bf not} get a printout.

\end{enumerate}

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\textsf{The user compares different Riemann sums and considers
        their accuracy.}
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