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\markright{Math 132, Midterm 2}

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\begin{center}

\textbf{College of the Holy Cross, Fall Semester, 2004} \\
\textbf{Math 132, Midterm 1} (All Sections) \\
\textbf{Thursday, March 31, 6 PM}
\end{center}

\vfil

\noindent\textbf{Name}: \underline{\phantom{Name Name Name Name Name}}
\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/\  16\hfil \Strut \\
\hline
2  & \hfil/\  12\hfil \Strut \\
\hline
3  & \hfil/\  18\hfil \Strut \\
\hline
4  & \hfil/\  14\hfil \Strut \\
\hline
5  & \hfil/\  14\hfil \Strut \\
\hline
6  & \hfil/\  14\hfil \Strut \\
\hline
7  & \hfil/\  12\hfil \Strut \\
\hline\hline
Total   & \hfil/100\hfil \Strut \\
\hline
\end{tabular}
\end{center}

\newpage

\begin{enumerate}
  \item The graph of the function $f(x)$ is shown below. 
  
  \begin{center}
    \input{exam2questions_3.eepic}
  \end{center}
  
  \begin{enumerate}
    \item{} [10 points] Use the graph of $f$ to 
    compute the following approximations of $\int_0^8 f(x)\,dx$.
    
    \item{} {\bf Solution:}  Since $n = 2$, $\Delta x = \frac{8 - 0}{2} = 4$.
    
    $LEFT(2):\underline{\ 28\ } = 0\cdot 4 + 7\cdot 4$
    \vfill
     $RIGHT(2):\underline{\ 68\ } = 7\cdot 4 + 10\cdot 4$
    \vfill
     $MID(2):\underline{\ 52 \ } = 4\cdot 4 + 9\cdot 4$
    \vfill
     $TRAP(2):\underline{\ 48\ } = \frac{1}{2}(LEFT(2) + RIGHT(2))$
    \vfill
     $SIMP(2):\underline{\ 50.6\ } = \frac{2 MID(2) + TRAP(2)}{3}$
    \vfill
    \item{} [6 points] For each method, decide whether the approximation 
    of $\int_0^8 f(x)\,dx$ is an {\it overestimate}, an {\it underestimate}, 
    or that this {\it cannot be determined} from the given information. 
    In the case of an overestimate or underestimate, briefly explain your 
    reasoning. 
    
    \begin{tabular}{|c|c|c|}
    \hline
    Method & Type of Estimate \qquad& Reason\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \\
    \hline
    $LEFT\begin{matrix}\ \\\ \end{matrix}$ &&\\
    \hline
    $RIGHT\begin{matrix}\ \\\ \end{matrix}$ &&\\
    \hline
    $MID\begin{matrix}\ \\\ \end{matrix}$ &&\\
    \hline
    $TRAP\begin{matrix}\ \\\ \end{matrix}$ &&\\
    \hline
    $SIMP\begin{matrix}\ \\\ \end{matrix}$&&\\
    \hline
    \end{tabular}
    
  \end{enumerate}
  
  
  \newpage
  
  \item 
  \begin{enumerate}
  \item{} [6 points] Set up an integral that represents the arclength of the portion of the 
  graph of $y=\sin(x)$ between $x=0$ and $x=\pi$. {\bf Do not evaluate the integral.} 
  \vfill
  \item{} [6 points] Approximate the integral in part (a) using a left hand sum with 
  $n=4$ subintervals.
  \vfill
  \end{enumerate}
  
  
  \newpage
  
  \item Rewrite each of the following improper integrals as a limit, or limits. 
  State whether the integral converges or diverges, and compute its value 
  if it converges.
  \begin{enumerate}
    \item{} [6 points] $\displaystyle\int_{-1}^2\frac1{x^3}\,dx$
    \vfill
    \item{} [6 points] $\displaystyle\int_0^2\frac1{\sqrt{4-x^2}}\,dx$
    \vfill
    \item{} [6 points] $\displaystyle\int_0^\infty e^{-4x}\,dx$
    \vfill
  \end{enumerate}
  
  
  \newpage
  \item Let $R$ denote the region bounded by $y=\sin x$, $y=\cos x$, $x=0$ and 
  $x=\pi/4$. 
  \ 
  \vskip 10pt
  \ 
  \begin{center}
    \input{exam2questions_2.eepic}
  \end{center}
  \begin{enumerate}
  
    \item{} [7 points] Find the area of $R$.
    \vfill
    \item{} [7 points] Find $\overline{x}$, 
    the $x$-coordinate of the center of mass of~$R$. (Assume the region 
    has constant mass density $\delta=1$.)
    \vfill
  \end{enumerate}
  
  \newpage
  
  \item Let $R$ be the region bounded by the curve $y=e^x$ and the lines $x=0$ and $y=e$. 
  
  \begin{center}
    \input{exam2questions_4.eepic}
  \end{center}
  
  \begin{enumerate}
    \item{} [8 points] Find the volume of the solid obtained by revolving $R$ about the $x$-axis.
    \vfill
    \item{} [6 points] Set up an integral that represents the volume of the solid whose base is $R$ and 
    whose cross-sections perpendicular to the $x$-axis are squares with base lying in $R$. 
    {\bf Do not evaluate the integral.}
    \vfill
  \end{enumerate} 
  
  \newpage
  
  \item Suppose a metal rod of length $2$ meters has a mass density 
  $\delta(x)=5+1.2x^2$ kg per meter. 
  \begin{enumerate}
    \item{} [7 points] Find the total mass of the rod.
    \vfill
    \item{} [7 points] Find the center of mass of the rod.
    \vfill
  \end{enumerate}
  
  \newpage
  
  \item A 200 meter tall pyramid has a square base with side length 100 meters. 
  Its mass density (in kg/m${}^3$) at height $h$ meters above the base is given by $\delta(h)=1+0.01h$. 
  
  \begin{center}
    \input{exam2questions_5.eepic}
  \end{center}
  \vskip 5pt
  \ 
  \begin{enumerate}
    \item{} [6 points] Write down a Riemann sum that approximates the total mass of the pyramid. 
    \vfill
    \item{} [6 points] Using the answer to part (a), write down an integral that equals the 
    mass of the pyramid. {\bf Do not evaluate the integral.}
  \vfill
  \end{enumerate}
  
\end{enumerate}
\end{document}


    \item $\displaystyle\int_0^\infty\frac{1}{\sqrt{x}(1+\sqrt{x})^2}\,dx$
   \item $\displaystyle\int_5^\infty\frac1{x^2-16}\,dx$
    \item $\displaystyle\int_0^1\ln x\,dx$.
    \item $\displaystyle\int_0^\infty e^{-x}\cos x\,dx$.