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\begin{center}

\textbf{College of the Holy Cross, Spring 2014} \\
\textbf{Math 136, section 1 -- Midterm Exam 2} \\
\textbf{Friday, March 28}
\end{center}

\vfil

\noindent\textbf{Your Name}: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$



\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/ 30\  \hfil \\
\hline
II  & \hfil/ 15\  \hfil \\
\hline
III  & \hfil/ 15\  \hfil \\
\hline
IV  & \hfil/ 30\  \hfil \\
\hline
V  & \hfil/ 10\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfil\eject

\noindent
I.  For these problems, you must show \emph{all work necessary} to justify your answers, 
but you may consult the portions of the table of integrals provided as needed.  If
you use a table entry, identify it by number.  
\begin{enumerate}
\item[A.] (15) Integrate with the partial fraction method:  
${\displaystyle \int \frac{3x + 1}{x^3 + 16x}\ dx}$

\vskip 7.5truein
\MyAnswer{Answer:}{$\phantom{A very very very very very very very long answer here}$}%
\vfill\eject

\item[B.] (15)  Integrate via a trigonometric substitution:
${\displaystyle \int x^3 \sqrt{81 - x^2}\ dx}$
\vskip 7.5truein
\MyAnswer{Answer:}{$\phantom{A very very very very very very very long answer here}$}%
\end{enumerate}
\vfill\eject

\noindent
II.   
\begin{enumerate}
\item[A.] (5) Use midpoint Riemann sums with $n = 2$ 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 $n = 2$ 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=1.5truein,height=1.5truein]{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
III.  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.]  (5) ${\displaystyle \int_1^\infty e^{-x}\ dx}$
\vskip 4.0truein
\MyAnswer{}{$\phantom{A very long answer here}$}%
\vfill\eject
\item[B.] (10) ${\displaystyle \int_{-1}^1 \frac{1}{x^2}\ dx}$
\vskip 4.0truein
\MyAnswer{}{$\phantom{A very long answer here}$}%
\end{enumerate}
\noindent
IV.  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}
\vfill\eject
\item[B.] (5)  Set up and compute an integral to find the area of $R$.
\vskip 1.9truein
\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 1.9truein
\MyAnswer{Volume $=$ }{$\phantom{A very long answer here}$}%
\item[D.] (10)  A solid has the region $R$ as base and cross-sections
by planes perpendicular to the $x$-axis are isosceles right triangles
with hypotenuse extending from the lower boundary to the upper boundary
of the region.  Set up and compute an integral to find the volume.  
\vskip 1.9truein
\MyAnswer{Volume $=$ }{$\phantom{A very long answer here}$}%
\end{enumerate}
\noindent
V. (10) Suppose that the region $R$ defined by $0 \le y \le f(x)$ and $a \le x \le b$
has area $A$ and lies above the $x$-axis.  When $R$ is rotated about the $x$-axis it 
sweeps out a solid with volume $V_1$.  When $R$ is rotated about the line $y = -k$, 
where $k > 0$, it sweeps out a solid with volume $V_2$.  Express $V_2$ in terms of 
$V_1, k, A$.  
\vskip 4.5truein 
\MyAnswer{Volume $V_2 =$ }{$\phantom{A very long answer here}$}%
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