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\title[\hskip 0.2inExam 2\hfill Name:\hskip 2in]{Exam 2:  Integral calculus}

\begin{document}

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\centerline{\Large{\sc{Month 0: Mathematics for Computer Science}}}
\medskip

\maketitle

\vfill


\centerline{\LARGE{September 17, 2000}}

\vskip 1in

\hskip 2in\Large{Name:}

\vskip 1in

\noindent You may consult your paper containing trigonometric
identities, but you may not consult any other books or papers.  You
may use a calculator (or the calculator on your computer), but you may
not use any other computing or graphing device other than your own
head! 

\vfill\pagebreak


\centerline{\sc Identities that you may need}

\vskip 0.3in


$$
\begin{array}{|c|c|}
\hline
1=\cos^2(\theta)+\sin^2(\theta) & \cos^2(\theta)=1-\sin^2(\theta)\\
\hline
\cot^2(\theta)=\csc^2(\theta)-1 & \sec^2(\theta)=1+\tan^2(\theta) \\
\hline
\end{array}
$$

If $S_r$ is a sphere of radius $r$, then the volume of $S$ is
$$
V(S_r)=\frac{4}{3}\pi r^3.
$$
If $C_{r,h}$ is a cylinder of radius $r$ and height $h$ with no lids, then
the surface area of $C_{r,h}$ is
$$
S.A.(C_{r,h})=2\pi rh,
$$
and the volume is
$$
V(C_{r,h})=\pi r^2h.
$$

\vfill\pagebreak



\begin{enumerate}

\item Take the following integrals.
\begin{enumerate}
\item $\int_0^2 4x^3-2x+1\  dx$
\vfill
\item $\int \tan(\theta)\ d\theta$
\vfill
\item $\int 3x^2\ln(x)\ dx$
\vfill\pagebreak
\item $\int \frac{4\sin(\theta)-5}{(\sin(\theta)-2)(\sin(\theta)-1)}
\cdot\cos(\theta)\ d\theta$
\vfill
\item $\int_0^{\frac{1}{2}} \frac{1}{1-x}\ dx$
\end{enumerate}

\vfill\pagebreak

\item Consider the functions $f(x)=x^2$ and $g(x)=x^3$.

\begin{enumerate}
\item Rotate the region between $f$ and $g$ from $x=0$ to
$x=1$ around the $y$-axis, and compute the volume of that solid of
revolution. 
\vfill 
\item  Rotate the region between $f$ and $g$ from $x=0$ to
$x=1$ around the $x$-axis, and compute the volume of that solid of
revolution. 
\end{enumerate}
\vfill\pagebreak

\item A horn shaped solid is formed by a moving circle perpendicular
to the $y$-axis whose diameter lies in the $xy$-plane and extends from 
$y=25x^2$ to $y=x^2$.  (See figure below for a rough sketch.)    Find
the volume of this solid between $y=0$ and $y=5$.


\begin{figure}[h]
\hfill\psfig{figure=horn.ps,width=3in}
\end{figure}

\vfill\pagebreak

\item ({\sc Short Answer})  Please answer the following questions in
full English sentences.  (Mathematicians have to write too!)

\begin{enumerate}
\item If you are given a function $f(x)$ as in the figure below, and
$a$ and $b$ are numbers between $A$ and $B$, what 
does $\int_a^b f(x)\ dx$ compute?
\begin{figure}[h]
\hfill\psfig{figure=function2.ps,width=2in}
\end{figure}
\vfill
\item Describe when and why $\int_a^b f(x)\ dx$ might be a negative number.
\end{enumerate}


\vfill\pagebreak

\noindent {\sc{Extra problems -- Some logic puzzles.}} Only do these
problems if you have completed and checked over your exam, and feel like
taking a look at these!  They will not count
towards your exam grade, but are meant to be fun problems to think
about!

\begin{enumerate}
\item ({\sc Pirates of the Caribbean}) There is a pirate ship of $10$
pirates, and they have found $100$ gold coins.  Their algorithm for
dividing up gold coins is the following.  They rank themselves from
strongest to weakest ($10$ being the strongest), and then the
strongest pirate (number $10$) suggests a plan for dividing up the
coins.  The pirates then vote on the plan, and if the plan gets aproved
by $50\%$ of the group, then that is how the coins are divided.  If
the plan is not approved, Pirate $10$ is thrown overboard (and
therefore drowns, since pirates cannot swim), and Pirate $9$ must now
propose a plan.  You are Pirate $10$.  What do you propose?  (You may
assume that you do not want to drown, and that the pirates will vote
for a plan that they feel will maximize their gold coin intake.)
\vfill\pagebreak
\item ({\sc Whistling in the dark}) You enter a house, and next to the 
door, there is a light switch with $3$ switches, all in the off
position.  You know that one of the switches controls a $60$ watt 
lightbulb in a closet on the $3^{\mathrm rd}$ floor, and you want to
determine which one.  You may flip the switches in any way you like,
and then you must go up to the closet, open the door, and immediately
deduce which switch controlled the light.  (No running up and down the 
stairs, and you may assume that the other two switches do not connect
to anything.)  What do you do?
\end{enumerate}


\end{enumerate}





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