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\centerline{Mathematics 136, section 2 -- Advanced Placement Calculus}
\centerline{Makeup Exam 3 -- December 4, 2009}
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\noindent
{\it Directions}
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Do questions I - V.  {\it You must show all work done to justify your answers in your solutions.  If you use an entry from the extract of the table of integrals, identify it
by number.}   There are 100 regular points and 10 Extra Credit points.
Do all work in the blue exam booklet.  {\it Good ``hunting!''}
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\begin{enumerate}
\item[I.]  Let $R$ be the region in the plane bounded by $y = 17 - x^2$ and 
$y = 1$.  
\begin{enumerate}
\item[A)] (5)  Sketch the region $R$.  
\item[B)] (5)  Find the {\it area} of $R$.
\item[C)] (10) Find the {\it volume} of the solid generated by rotating $R$ about the
$x$-axis.
\item[D)] (5)  Set up, but do not evaluate, the integral or integrals to compute the 
{\it volume} of the solid generated by rotating $R$ about the $y$-axis.  
\end{enumerate}
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\item[II.] 
\begin{enumerate}
\item[A)] (10) Integrate by trigonometric substitution 
(not the table) $\int \frac{1}{\sqrt{9 - x^2}}\ dx$. 
\item[B)] (10) Does the integral $\int_0^3 \frac{1}{\sqrt{9 - x^2}}\ dx$ converge?
If so, find its value.  (For this part, you can use an appropriate table entry if
you were not able to get an answer for A.)
\end{enumerate} 
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\item[III.] (15)  A water tank has the shape of an inverted cone
(that is, the ``point'' of the cone is at the bottom) with
height $8$ meters and radius $4$ meters.  If the tank is
full of oil (density 1200kg per cubic meter), find the {\it work} done in
pumping all the water out the top of the tank.  The acceleration of 
gravity is $9.8$ meters per second squared.  
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\item[IV.]  (15)  Set up and evaluate the integral to compute the {\it arclength}  
of $y = \ln(\cos(x))$, $0 \le x \le \pi/4$.   (Use the table to evaluate.)
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\item[V.]  All parts of this question refer to the differential equation
\[
\frac{dy}{dx} = (y^2 + 1)/4
\]
\begin{enumerate}
\item[A)] (10)  Sketch the {\it slope field} of this equation, showing 
the slopes at points on the lines $y = 0,1,2,3,4,5$, and $-3 \le x \le 3$.
\item[B)]  (5) On your slope field, sketch the graph of the 
{\it solution} of the equation with $y(0) = 1$.
\item[C)]  (10) This is a separable equation; find an explicit formula
for the {\it solution} satisfying the initial condition $y(0) = 1$. 
\end{enumerate}
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\item[] {\it Extra Credit} (10)  Set up the integrals to 
find the coordinates of the {\it centroid} of the triangle
with corners at $(0,0)$, $(4,0)$, and $(4,7)$.  
You do not need to evaluate.

\end{enumerate}
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