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\title{Math 1b. Approximate Integration}

\date{Spring 2006}

\begin{document}

\maketitle





\begin{enumerate}

\item 
Consider $I = \int_0^1 \sin( x^2) \, dx$.
\begin{enumerate}

\item
Complete the following table.

\bigskip


\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
$x$ & 0.0625 & 0.125 & 0.1875 & 0.25  \\  \hline
$\sin( x^2)$ &  0.00390624 &  0.01562436  &  0.03514901  &  \hspace*{5em}    \\ \hline
$x$ & 0.3125 & 0.375 & 0.4375 & 0.5  \\  \hline
$\sin( x^2)$ &  0.09750110 &  \hspace*{5em}  &  0.19023965   &  \hspace*{5em}  \\ \hline
$x$ & 0.5625 & 0.625 & 0.6875 & 0.75 \\  \hline
$\sin( x^2)$ &  0.31115322 &  \hspace*{5em}  &  0.45525291    &  \hspace*{5em}  \\ \hline
$x$ & 0.8125 & 0.875 & 0.9375 & 1.0 \\  \hline
$\sin( x^2)$ & 0.61324028 &  \hspace*{5em}  & 0.77004153    &  \hspace*{5em}  \\ \hline\end{tabular}
\end{center}


\bigskip

\item
If $n = 8$, use a Left Sum  to estimate $I$.

\vfill

\item
If $n = 8$, use a Right Sum  to estimate $I$.

\vfill

\item
If $n = 8$, use the Midpoint Rule to estimate $I$.

\vfill

\item
If $n = 8$, use the Trapezoid Rule to estimate $I$.

\vfill

\item
If $n = 8$, use Simpson's Rule to estimate $I$.

\vfill

\end{enumerate}

\clearpage

\item 
Let  $I = \int_0^1 \sin( x^2) \, dx$.
\begin{enumerate}

\item
How many subdivisions are needed to assure that the error is less than $10^{-6}$ if we use the Midpoint Rule to estimate $I$?

\vfill

\item
How many subdivisions are needed to assure that the error is less than $10^{-6}$ if we use the Trapezoid Rule to estimate $I$?

\vfill

\item
How many subdivisions are needed to assure that the error is less than $10^{-6}$ if we use Simpson's Rule to estimate $I$?

\vfill

\end{enumerate}

\end{enumerate}



 \end{document}