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\begin{document}

\begin{center}

\textbf{College of the Holy Cross, Fall Semester, 2004} \\
\textbf{Math 132, Midterm 1} (All Sections) \\
\textbf{Wednesday, February~23, 6 PM}
\end{center}

\vfil

\noindent\textbf{Name}: \Blank
\vskip 0pt plus .3fil

\paragraph{Instructions} Please write your answers in the spaces
provided, and show work on the test itself. Use the back of the
preceding page if you need more space for scratch work.

\vfil

\noindent\textbf{Please do not write in the space below}

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss    \Strut \\
\hline\hline
1  & \hfil/\  20\hfil \Strut \\
\hline
2  & \hfil/\  10\hfil \Strut \\
\hline
3  & \hfil/\  10\hfil \Strut \\
\hline
4  & \hfil/\  20\hfil \Strut \\
\hline
5  & \hfil/\  20\hfil \Strut \\
\hline
6  & \hfil/\  10\hfil \Strut \\
\hline
7  & \hfil/\  10\hfil \Strut \\
\hline\hline
Total   & \hfil/100\hfil \Strut \\
\hline
\end{tabular}
\end{center}

\vfil\eject

\begin{enumerate}
  
\item{} [10~points each]\quad Determine whether each of the following
  improper integrals converges, and if so, find its value.

  (a) $\ds\int_0^\infty e^{-4x}\,dx$
  \vfil\sep

  (b) $\ds\int_0^1 \frac{dx}{\sqrt{1-x}}$

  \vfil\eject

\item{} [10~points each]\quad Each part of this question concerns the
  region~$R$ in the plane bounded by  the graph
  $y=\frac{1}{6}x^3+\frac{1}{2}x^{-1}$, the $x$-axis, and the lines $x=1$,
  $x=4$.

  (a) Find~$\bar{x}$, the $x$~coordinate of the center of mass of~$R$,
  and mark it in the figure below.
  \vfil\input{midterm2a.eepic}\Answer{$\bar{x}=$}
  \sep

  (b) Find the volume of the solid generated when~$R$ is revolved
  about the $x$-axis.

  \vfil\vfil\vfil\Answer{Volume:}

  \noindent\textbf{Continued next page}
  \eject

  (c) Find the arc length of the curve along the top of~$R$.

  \vfil\Answer{Arc length:}
  \eject
  
\item{} [5~points each]\quad Each part concerns the integral
  $\ds\int_0^3\Bigl(2-\frac{1}{x-4}\Bigr)\,dx$.
  \begin{center}
    \input{midterm2b.eepic}
  \end{center}

  (a) Compute $\mathrm{LEFT}(3)$~and $\mathrm{RIGHT}(3)$.

  \vfil\Answer{$\mathrm{LEFT}(3)$}\Answer{$\mathrm{RIGHT}(3)$}
  \sep

  (b) Compute $\mathrm{TRAP}(3)$~and $\mathrm{MID}(3)$.

  \vfil\Answer{$\mathrm{TRAP}(3)$}\Answer{$\mathrm{MID}(3)$}
  \sep

  (c) Compute the exact value of the integral.

  \vfil\Answer{Exact value:}
  \sep

  (d) Use parts (b)~and (c) to compute the error for the trapezoid and
  midpoint formulas with three subintervals.

  \vfil\Answer{Trapezoid error:}\Answer{Midpoint error:}
  \eject

\ee
\end{document}

\item{} [10~points]\quad 

  \eject
  
\item{} [10~points each]\quad Find the integrals. Use the indicated
  method to get started.

  (a) $\ds\int_1^4\frac{\sqrt{1+\sqrt{x}}}{\sqrt{x}}\,dx$
  (Substitution)
  \vfil\sep

  (b) $\ds\int t\sec^2 t\,dt$ (Integrate by parts)
  \vfil\eject

\item{} [10~points each]\quad Compute the integrals

  (a) $\ds\int\frac{dx}{(4-x^2)^{3/2}}$
  \vfil\sep

  (b) $\ds\int\frac{dx}{x^2+4x+8}$
  \vfil\eject

\item{} [10~points]\quad Find the indefinite integral
  $\ds\int\frac{x^2+1}{x(x+2)(x-1)}\,dx$
  \vfil\eject
  
\item{} [10~points]\quad Compute any \textbf{two} of the following;
  clearly mark your choices and indicate the method of integration.
  \[
  \mathrm{(i)}\ \int\frac{1}{\sqrt{1-x^4}}\,dx\qquad
  \mathrm{(ii)}\ \int\frac{x}{\sqrt{1-x^4}}\,dx\qquad
  \mathrm{(iii)}\ \int\frac{x^2}{\sqrt{1-x^4}}\,dx\qquad
  \mathrm{(iv)}\ \int\frac{x^3}{\sqrt{1-x^4}}\,dx
  \]

\vfil

\end{enumerate}

\end{document}
