
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,epsfig}
\newcommand{\ds}{\displaystyle}
\newcommand{\vp}{$\vphantom{\big|}$}

\newcommand{\R}{\textbf{R}}
\newcommand{\Item}{\item{$\Box$}\quad}

\newcommand{\Answer}[1]{$\phantom{.}$\hfill
        #1\quad\fboxsep 20pt \fbox{$\phantom{\Big|Answer}$}} 

\newcommand{\MyTallAnswer}[2]{$\phantom{.}$\hfill
        #1\quad\fboxsep 20pt\fbox{$\vphantom{\Big|}#2$}}

\newcommand{\MyAnswer}[2]{$\phantom{.}$\hfill
        #1\quad\fboxsep 5pt\fbox{$\vphantom{\Big|}#2$}}

\newcommand{\soln}{\paragraph{Solution}}
\newcommand{\sep}{{\medskip\hrule\bigskip}}
\newcommand{\blank}{\underline{$\phantom{A blank}$}}

\setlength{\textwidth}{6.5in}
\setlength{\topmargin}{-0.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{9.0in}
\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Spring 2008} \\
\textbf{Math 132, Midterm Exam 2} (All Sections) \\
\textbf{Wednesday, March 26, 6 PM}
\end{center}

\vfil

\noindent\textbf{Your Name}: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\vskip 0pt plus .3fil
\noindent\textbf{Your Section}: 

\vskip 15pt
\leftline{Little (8:00am) $\underline{\qquad}$ Ballantine (9:00am) $\underline{\qquad}$} \leftline{DeStefano (10:00am) $\underline{\qquad}$ DeStefano (noon) $\underline{\qquad}$}

\vskip 30pt
\noindent\textbf{Instructions}:  For full credit, you must show 
{\it all work} on the test pages and place your final answer
in the box provided for the problem.  Use the back of the
preceding page if you need more space for scratch work.  The numbers
next to each part of the questions are their point values.

\vfil

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

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss \\
\hline\hline
I  & \hfil/ 15\  \hfil \\
\hline
II  & \hfil/ 25\  \hfil \\
\hline
III  & \hfil/ 30\  \hfil \\
\hline
IV  & \hfil/ 15\  \hfil \\
\hline
V  & \hfil/ 15\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfil\eject

\noindent
I.   
\begin{enumerate}
\item[A.] (5) Use midpoint Riemann sums with $n = 4$ to approximate $\displaystyle \int_1^2 \sqrt{1 + \ln(x)}\ dx$.  

\vskip 3.0truein
\MyAnswer{Midpoint approximation: }{$\phantom{A very long answer here}$}%
\item[B.] (5) Use the trapezoidal rule with $n = 4$ to approximate $\displaystyle \int_1^2 \sqrt{1 + \ln(x)}\ dx$. 

\vskip 3.0truein
\MyAnswer{Trapezoidal rule approximation: }{$\phantom{A very long answer here}$}%
\vfill\eject
\item[C.] (5) The function $y = \sqrt{1 + \ln(x)}$ is plotted below.
\begin{center}
\includegraphics[width=2.3truein,height=2.3truein]{132P2I.ps} 
\end{center}
\noindent
Check the appropriate boxes:
\vskip 10pt
\item[]The midpoint approximation is a overestimate $\Box$/underestimate $\Box$.
\item[]The trapezoidal rule approximation is a overestimate $\Box$/underestimate $\Box$.
\end{enumerate}

\vskip 10pt
\noindent
II.  For each of the following improper integrals, set up and evaluate the appropriate limits
to determine whether the integral converges.  If so, find its value; if not, say ``does not converge.''
(Credit will be given only for the correct limit calculation.)
\begin{enumerate}
\item[A.]  (8) ${\displaystyle \int_1^\infty e^{-x}\ dx}$
\vskip 3.0truein
\MyAnswer{}{$\phantom{A very long answer here}$}%
\vfill\eject
\item[B.] (8) ${\displaystyle \int_{-1}^1 \frac{1}{x^2}\ dx}$
\vskip 3.0truein
\MyAnswer{}{$\phantom{A very long answer here}$}%
\item[C.] (9) $\displaystyle \int_0^1 x\ln(x)\ dx$.  
\vskip 3.5truein
\MyAnswer{}{$\phantom{A very long answer here}$}%
\end{enumerate}
\vfil\eject
\noindent
III.  The region $R$ is bounded by the graphs $y = \sqrt{x}$ and $y = x^2$.  
\begin{enumerate}
\item[A.] (5)  Sketch the region $R$.
\begin{center}
\includegraphics[width=2.0truein,height=2.0truein]{132P2II.ps} 
\end{center}
\item[B.] (5)  Set up and compute an integral to find the area of $R$.
\vskip 2.0truein
\MyAnswer{Area $=$ }{$\phantom{A very long answer here}$}%
\item[C.] (10)  The region $R$ is rotated about the $x$-axis to generate
a solid.  Set up and compute an integral to find its volume.
\vskip 2.5truein
\MyAnswer{Volume $=$ }{$\phantom{A very long answer here}$}%
\item[D.] (10)  The region $R$ is rotated about the $y$-axis to generate
a solid.  Set up and compute an integral to find its volume.
\vskip 3.5truein
\MyAnswer{Volume $=$ }{$\phantom{A very long answer here}$}%
\end{enumerate}
(go to next page)
\vfill\eject
\noindent
IV. (15) Set up and compute the integral for the arclength of the curve
$x = 5t^2 + 2, y = t + 1$, $0 \le t \le 2$.  (You may consult the 
table of integrals for this one; if you do, say which table entry you are using.)
\vfill\eject
\noindent
V. (15) Find the average value of $\displaystyle{f(x) = \frac{x^3}{\sqrt{25 - x^2}}}$ on 
the interval $0 \le x \le 2$.  (Use a trigonometric substitution to compute 
the integral you need.)

\end{document}

