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\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Spring 2018} \\
\textbf{Math 134 Midterm Exam 2} \\
\textbf{Friday, March 16}
\end{center}

\vskip 100pt

\noindent\textbf{Your Name}: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\vskip 200pt


\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.
The following power sum formulae may be useful:

\vskip 30pt


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

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss \\
\hline\hline
I  & \hfil/ 40\  \hfil \\
\hline
II  & \hfil/ 45\  \hfil \\
\hline
III  & \hfil/ 15\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfil\eject
\noindent
I.  All parts of this problem refer to the region $R$ bounded by $y = x$, $y = x^2 + 1$, $x = 0$ and $x = 2$.  
\begin{enumerate}
\item[A.] (10)  Sketch the region $R$.
\vskip 3.0truein


\item[B.] (10) Compute the area of the region $R$.   
\vskip 3.0truein
\MyAnswer{Area $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}

\vfil\eject
\begin{enumerate}
\item[C.] (10) Compute the volume of the solid of revolution obtained by rotating the region $R$ about 
the $x$-axis.  
\vskip 3.0truein
\MyAnswer{Volume $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[D.] (10) Compute the volume of the solid of revolution obtained by rotating the region $R$ about
the line $x = -2$.  (Note: Any correct method is OK here.)
\vskip 3.0truein
\MyAnswer{Volume $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\noindent
II.  Compute each of the following integrals by an appropriate method (e.g. $u$-substitution, parts, etc.)
\begin{enumerate}
\item[A.] (15) $\displaystyle\int x^2 \cos(5x^3)\ dx$
\vskip 3.0truein
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[B.] (15) $\displaystyle\int x^2 \cos(5x) \ dx$
\vskip 3.0truein
\MyAnswer{Integral $=$}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
\vfill\eject
\begin{enumerate}
\item[C.]  (15) $\displaystyle \int e^{\sqrt{x}}\ dx$  (Hint: Let $u = x^{1/2}$.)
\end{enumerate}
\vfill\eject
\noindent
III. 
\begin{enumerate}
\item[A.] (10) Identify a $u$ and a $dv$, then carry out an integration by parts to derive the
following reduction formula for exponents $k \ge 1$.  
$$\int (\ln(x))^k\ dx = x(\ln(x))^k - k \int (\ln(x))^{k-1}\ dx$$
\vskip 3.0truein

\item[B.]  (5) Apply the formula in part A to integrate $\displaystyle \int (\ln(x))^2\ dx$.  (Note:  You 
can do this part even if you didn't see how to finish part A.)

\vskip 150pt
\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
 
\end{enumerate}
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\vskip 200pt 
\MyAnswer{Integral $ = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{enumerate}
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