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\hskip36pc \= Math.\ 301-02\\
\>Summer, 2005\\
\>R. B. Kearfott
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\bf Second Examination \\
\it Wednesday, June 29, 2005
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\noindent {\small {\bf Instructions:} This exam should be done on
your own paper. Your name should be on each sheet and on the back
of the last sheet; the answers should appear written carefully and
in order. If in doubt, show intermediate steps: Full credit may
not be given, even for correct answers, unless work is arranged
clearly and explained. This exam is closed book. You may leave
after handing in your exam paper, but be sure to check your
answers carefully. You may keep this exam sheet. Each entire
problem is worth 20 points.}

\begin{enumerate}
   \item  State whether the following improper integrals converge
          or diverge.  Give a reason for each of your answers.
          Evaluate those improper integrals that can be evaluated
          without resorting to numerical techniques.
          $$\begin{array}{rlrlrl}
             \mbox{(a)} & \displaystyle
               \int_{x=1}^\infty \frac{dx}{x^{1.001}} & \qquad
             \mbox{(b)} & \displaystyle
               \int_{x=0}^1 \frac{dx}{x^{1.001}} & \qquad
             \mbox{(c)} & \displaystyle
               \int_{x=0}^{\infty} \frac{e^{-x}}{\sqrt{x}}\/dx
          \end{array}$$
   \item Find the volume of the following figure by writing down
         an integral, then evaluating it.\newline
         \hskip1in{\mbox{\includegraphics[scale=1.0]{test_2_fig_1.eps}}}
   \item Compute the volume of the solid obtained by rotating the
         portion of the curve $y=\sqrt{x+1}$ between $x=0$ and
         $x=1$ about the line $y=0$.
   \item Compute the length of the portion of the curve $y=x\sqrt{x}$
         between $x=0$ and $x=1$.
   \item Compute the center of mass of an object extending from $x=0$
         meters to $x=10$ meters, and with density at position $x$
         equal to $e^{-x}$ kilograms per meter.
\end{enumerate}
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