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{\Large {\bf 
Mathematics 1210 \hspace{5ex}  Prof. Ken Golden\hspace{5ex} \\

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PRACTICE FINAL EXAM 
\hspace{5ex}Fall 2009 \\}

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Problems From Varberg, {\it et al.} (Chapter.Section.Number)
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%{\Large {\bf ANSWER KEY}}

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\begin{enumerate}

\item[1.] 
5.1.6. Use the three step procedure (slice, approximate, integrate)
to set up an integral or integrals for the area of the indicated
region. The region lies above the parabola $y=x^2-2$
and below the line $y=x+4$, and is enclosed by these two graphs.


\item[2.] 
5.2.17. Find the volume
of the solid generated by revolving about the  $x$-axis
the region bounded by the upper half of the ellipse
$$\frac{x^2}{a^2} +  \frac{y^2}{b^2} = 1$$
and the $x$-axis, and thus find the volume of a {\it prolate spheroid. }
Here $a$ and $b$ are positive constants, with $a>b$.


\item[3.] 
5.3.2. Find the volume of the solid generated when the region $R$
bounded by the curves $y=x^2$, $x=1$, $y=0$ is revolved about the
$y$-axis.
Do this by performing the following steps:
(a) Sketch the region $R$.
(b) Show a typical rectangular slice properly labeled.
(c) Write a formula for the approximate volume of the shell
generated by this slice.
(d) Set up the corresponding integral.
(e) Evaluate this integral.


\item[4.] 
5.4.10. Sketch the graph of the parametric equation
$$x=\sqrt 5 \sin {2t} -2 \quad  \quad y=\sqrt 5 \cos {2t} - \sqrt 3 ;
\quad \quad 0 \leq t \leq \frac{\pi}{4} ,$$
and find its length.


\item[5.] 
5.4.21a. Find the length of the curve
$$y=\int_1^x \sqrt{ u^3 -1 }\; du, \quad 1 \leq x \leq 2 .$$

\item[6.] 
5.5.6. For a certain type of nonlinear spring, the force required
to keep the spring stretched a distance $s$ is given by the formula
$F=ks^{4/3}$. If the force required to keep it stretched 8 inches is
2 pounds, how much work is done in stretching this spring 27 inches?

\item[7.] 
5.5.22. According to Coulomb's Law, two like electrical charges
repel each other with a force that is inversely proportional to
the square of the distance between them. If the force of repulsion
is $10$ dynes ($1$ dyne = $10^{-5}$ newton) when they are $2$ centimeters
apart, find the work done in bringing the charges from $5$ centimeters
apart to $1$ centimeter apart.

\item[8.] 
5.6.25.  Use Pappus' Theorem to find the volume of the solid obtained
when the region bounded by $y=x^3$, $y=0$, and $x=1$ is revolved
about the $y$-axis (see Problem 11 for the centroid). Do the
same problem by the method of cylindrical shells to check your
answer.


\item[9.]
Find the temperature profile 
$T(x)$ on the interval $[0,1]$ satisfying the 
steady state heat equation $$\ds \frac{d^2 T}{dx^2}=0$$
and the boundary conditions $T(0)=4$ and $T(1)=2$.

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\end{enumerate}

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