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\centerline{MATH 136 -- Calculus 2}
\centerline{Discussion Day on Volumes by Cavalieri's Principle}
\centerline{September 28, 2016}
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\noindent
{\it Background}
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Recall \emph{Cavalieri's Principle}:  If we have a solid object ``lined up'' along 
the $x$-axis as a reference line,  
\begin{itemize}
\item the solid extends from $x = a$ to $x = b$, and
\item the \emph{area of the cross-section} of the solid at a general location $x$
is given by a known function $A(x)$, then
\end{itemize}
the volume of the solid is computed by the definite integral of the cross-section
area function:
$$V = \int_a^b A(x)\ dx.$$
Today, we will practice using this on some interesting examples.
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\noindent
{\it Questions}
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\begin{enumerate}
\item[(1)] First a ``thought question'' (no calculations!)  Suppose we have an ``oblique''
circular cylinder of radius $r$, where the axis (the ``center line'') of the cylinder meets 
the plane of each of the circular cross-sections at an angle $\theta \ne \frac{\pi}{2}$.
Draw such a cylinder and determine its volume, if the height (the vertical distance between the top and 
the bottom) is $h$  Explain how you can tell by using Cavalieri's Principle.
\item[(2)]  Given:  The area enclosed by an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is 
$\pi ab$.  Using this fact, find the volume of a right cone of height $h$ whose base is the 
ellipse $\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$.  Note:  $A,B$ are given values here.  The formula
for the area enclosed by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ works no matter what $a$ and $b$ are.  
\item[(3)] A plane inclined at an angle of $\frac{\pi}{4}$ passes through a diameter of the base of 
a right circular cylinder of radius $R$.  Find the volume of the region within the cylinder and below the plane.
There is a picture in problem 20 on page 349 of our textbook that may help you visualize this and
find a good way to slice it and apply Cavalieri.
\item[(4)]  The base of a solid is the triangle with vertices at $(0,0), (1,0), (0,1)$.  The
slices by planes perpendicular to the $x$-axis are semicircles with diameters extending from
the $x$-axis up to the point on the line through $(1,0)$ and $(0,1)$ with that $x$-coordinate.
What is the volume of this solid?
\end{enumerate}
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