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\centerline{MATH 136 -- Calculus 2}
\centerline{Volumes by Slicing}
\centerline{March 10, 2020}
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\noindent
{\it Background}
If a solid extends along the $x$-axis between $a$ and $b$ and the
area of the cross-section by a plane $x = {\rm const}$ is given by some
function $A(x)$ for all $x$, then {\it Cavalieri's Principle} (named after
the Italian Renaissance mathematician Bonaventura Cavalieri, 1598-1647)
says that 
$${\rm Volume} = \int_a^b A(x)\ dx$$
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\noindent
{\it Questions}
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\begin{enumerate}
\item A circular cone extends from $x = 0$ to $x = 4$ along the $x$-axis.
The cross-section in each plane $x = {\rm constant}$ is a circle 
whose radius increases linearly from $r = 0$ 
at $x = 0$ to $r = 5$ by the time $x$ reaches $4$.  
\begin{enumerate}
\item Write the radius $r$ as a function of $x$ on the interval $0$ to $4$.
\item Write the area of the circular cross-section as a function of $x$.
\item Find the volume by applying the Cavalieri Principle equation above.
\item Check your result with the formula for the volume of a cone from 
high school geometry.
\end{enumerate}
\item A solid extends along the $x$-axis from $x = -3$ to $x = 3$.
The cross-section in each plane $x = {\rm constant}$ is a semicircle
with radius $e^x$.  Find the volume of the solid using Cavalieri's Principle.
\item A solid extends from $x = 0$ to $x = 10$ along the $x$-axis.  
The cross-section at $x$ is a square with side $\sqrt[4]{25 + x^2}$.
Set up the integral to compute the volume using Cavalieri's Principle.
Evaluate the integral using a trigonometric substitution or a table of
integrals.  
\item In the previous problem, would it make an difference if the 
squares all had center point along the $x$-axis or if all the squares
had one corner at a point on the $x$-axis?
\end{enumerate}
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