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\def\header{{\hfill Exam 2 \hfill A. Miller \hfill Spring 2001
                                       \hfill Math 217\hfill}}
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\begin{document}

\bigskip

Show all work. {\bf Circle your answer.}

No books, no notes, no calculator, no cell phones, no pagers, no
electronic devices at all.

Solutions will be posted shortly after the exam:
www.math.wisc.edu/$\sim$miller/m217

\par\vskip .25in
\begin{center}
Name\makebox[3in]{\hrulefill} \\
\end{center}

\begin{center}
{\bf Hand in to your TA -- Circle your DIScussion section (column one):} \\
\end{center}
\begin{center}
\begin{tabular}{l|l|l|l}
DIS 321 & 7:45   TR  & 901  Van Vleck & Forray, Susan Jane     \\
DIS 322 & 8:50   TR  & 6224 Soc Sci   & Bisgard, James William \\
DIS 324 & 9:55   TR  & 123  Ingraham  & Bisgard, James William \\
DIS 326 & 11:00  TR  & B235 Van Vleck & Leete, Jennifer A.     \\
DIS 327 & 11:00  TR  & 113  Ingraham  & Lu, Tianshi            \\
DIS 328 & 12:05p TR  & B235 Van Vleck & Leete, Jennifer A.     \\
DIS 329 & 12:05p TR  & 6228 Soc Sci   & Lu, Tianshi            \\
DIS 330 & 1:20p  TR  & B321 Van Vleck & Forray, Susan Jane     \\
DIS 332 & 2:25p  TR  & B135 Van Vleck & Hur, Youngmi           \\
DIS 334 & 3:30p  TR  & B329 Van Vleck & Hur, Youngmi           \\
\end{tabular}
\end{center}





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\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Problem & Points & Score \\  \hline\hline
  \input{exam.tbl}
  \hline \hline
 \end{tabular}
\end{center}

\immediate\openout\examTBLFile=exam.tbl

\setcounter{page}{0}

\prob{10} % 5.2-10
The area bounded by the curves $y=x^3$, $y=0$, and $x=c>0$
has area $1$. What is $c$?
\answer $c=\sqrt{2}$

\prob{10} % 5.4-14
The base of a certain solid is the region bounded by
the curves $y=x^2$ and $y=1$.  Each plane section of the
solid cut out by a plane perpendicular to the $x$-axis is
a square with one edge of the square in the base of the
solid.   Find the volume of the solid.
\answer $16\over 15$


\prob{10} % 5.5-18
The area bounded by the curves $y=\sqrt{x}$ and $y=x$
is rotated about the $y$-axis.  Find the volume.
\answer $2\pi\over 15$

\prob{10} % 5.7-3
Find the lenght of the curve $x^2=4y^3$ from $(0,0)$ to $(2,1)$.
\answer ${2\over 27}(10^{3/2}-1)$


\prob{10} % 5.8-2
Find the surface area of the surface generated by rotating about the
line $y=1$ the arc of the curve $y={{x^3}\over 3}+1$
between $x=0$ and $x=1$.
\answer ${\pi\over 9}(2^{3/2}-1)$


\prob{10}  % 4.8
Prove the following part of the fundamental theorem of calculus:
\par \bigskip
Suppose $f(x)$ is continuous on the interval $[a,b]$,
$F(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$ with
the $F^\prime(x)=f(x)$ for all $x$ in $(a,b)$.  Let
$n$ be a large integer.
\par Define $\triangle x={{b-a}\over n}$.
\par Define $x_k=a+k\triangle x$ for $k=0,1,\ldots,n$.
\par\bigskip
  Show there exists $c_k$ with $x_{k-1}\leq c_k\leq x_k$
for $k=1,2,\ldots,n$ and such that
$$F(b)-F(a)=\sum_{k=1}^n \; f(c_k)\triangle x$$
\answer See Exam 1 problem 6.

\finishtbl
\printanswers

\end{document}