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\title{Math 1a.  Lecture 9 \\
Volumes}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to calculate volumes by integrating over cross-sectional area.

\item
To understand and be able to calculate volumes of solids of revolution using\footnote{We will not cover the method of cylindrical shells.}
$$
V = \int_a^b \pi [f(x)]^2 \, dx.
$$

\end{itemize} 

\section{Some Volumes that We Know}

\begin{itemize}

\item
Rectangular Box: $V = lwh$

\item
Cylinder: $V = \pi r^2 h$

\item
Cone: $V = \displaystyle\frac{1}{3} \pi r^2 h$

\item
Sphere: $V = \displaystyle\frac{4}{3} \pi r^3$

\end{itemize}



\section{Definition of Volume}

We can approximate the volume of a solid if we know the area of a cross-section at each point.  If $S$ is a solid lying between $a$ and $b$ and $A(x)$ is the cross-sectional area of $S$ on a the plane passing through $x$ and perpendicular to the $x$-axis, then
$$
V = \sum_{i=1}^n V_i \approx \sum_{i = 1}^n A(x_i^*) \Delta x_i.
$$
We define volume to be 
$$
V = \sum_{i=1}^n V_i = \lim_{n\to \infty}\sum_{i = 1}^n A(x_i^*) \Delta x_i = \int_a^b f(x) \, dx.
$$

\section{The Volume of a Sphere}

Show that the volume of a sphere of radius $r$ by rotating $y = \sqrt{r^2 - x^2}$ about the $x$-axis is $(4/3) \pi r^3$.  Note that this leads to finding the volumes of solids of revolution
$$
V = \int_a^b \pi [f(x)]^2 \, dx.
$$
where $f$ is rotated about the $x$-axis on the interval $a \leq x \leq b$.


\section{Worksheet Problems}

\begin{enumerate}

\item 
Find the volume of a sphere of radius $r$ by rotating $y = \sqrt{r^2 - x^2}$ about the $x$-axis.



\item 
Find the volume obtained by rotating the region bounded by $y = x^3$ and $y = x$ for $x \geq 0$ about the $x$-axis.



\item 
Find the volume obtained by rotating the region bounded by $y = \sqrt{x}$ and $y = x$  about the line $y = 1$.



\item
The base of a solid object $S$ is the region bounded by the parabola $y^2 = 3x$ and the line $x = 3$.   Cross-sections for the object perpendicular to the $y$-axis are equilateral triangles.  What is the volume of $S$?


\end{enumerate}


\section*{References}

\begin{itemize}

\item
\S 6.2 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.


\end{itemize}



\section*{Notes}

\subsection*{March 1, 2006}



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