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Math 132: Calculus for Physical and Life Sciences 2 \\ 
Answers for Problem Set 6 \\ 
Due Friday, March 14, 2008, at the beginning of class.
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\begin{enumerate}
\item[1.] 
\begin{enumerate}

\item[(a)] Cross sections are disks with radius $x^2 + 2$.  
Volume of solid is $\frac{293\pi}{15}$.

\item[(c)]  The cross-section by a plane $x = c$ is a washer with inner radius
$2 - \sin(x)$ and outer radius $2$.  The volume is  $8\pi - \frac{\pi^2}{2}$.


\item[(e)] 
The solid generated rotating around the $x$-axis is the union of two solids meeting
only along the plane $x = \sqrt{2}/2$.  The cross-sections of the first are
the washers with inner radius $x$ and outer radius $2x$.  The cross-sections
of the second are also washers with inner radius $x$ and outer radius
$1/x$.  The 
total volume is $\frac{4\pi}{3}(\sqrt{2}-1)$.

\item[(g)]  The cross-section by a plane $y = d$ is a washer with inner radius
$\frac{1}{y}$ and outer radius 2.  The volume equals $\frac{49\pi}{4}$.
\end{enumerate}

\item[2.] {\it Answer:}  $36\sqrt{3}$.


\item[3.] {\it Answer:} $40000\pi$ cubic feet.  
  
\item[4.]

\begin{enumerate}
\item[(a)] 
The graph is a straight line from the point $(3,-2)$ to the point $(11,4)$.
The arclength is $10$.

\item[(c)]
The curve is the arc of a circle with radius 3 and center $(0,0)$ starting
at $(0,3)$ and proceeding clockwise to $(0,-3)$ (one half of the full circle).
The arclength is $3\pi$.
\end{enumerate}

\item[5.] Arclength $\doteq 3.343018$.

\item[6.] 

\begin{enumerate}
\item[(a)] {\it Answer:} $\frac{-2}{9\pi}$.
\end{enumerate}

\item[8.] {\it Answer:}  $3.44$ units.
\end{enumerate}
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