\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{amssymb}
\textwidth = 6.5 in
\textheight = 9 in
\oddsidemargin = 0.0 in
\evensidemargin = 0.0 in
\topmargin =0 in
\headheight = 0.0 in
\headsep = 0.0 in
\parskip = 0.2in
\parindent = 0.0in

\begin{document}



\large 
\underline
{MA2160 \hspace{5mm}Calculus II with Technology \hspace{50mm} Ch. 8-2}

Again applying the principles of Riemann sum, we can find volumes of various kinds of looking shapes when revolving areas around the $x$- or $y$-axis.

\Large
Goal
\large
\begin{enumerate}
\item Identify three shapes we can use to approximate the volume of a slice of volumes of revolution and what circumstances we use each one.
\item Apply a definite integral to the formulas of these shapes to find the exact volume.
\item Find the length of the arc of an interval of a function.
\end{enumerate}

When we take an area and revolve it around either axis, we have methods of finding this volume of revolution.  If the revolution is around the $x$-axis (or any horizontal line), we can use the shape of either a circle or washer to model one slice of the volume.  If the revolution is around the $y$-axis (or other vertical line), we can use the shape of a cylindrical shell.

\begin{center}
Steps to find volumes of revolution
\end{center}
\begin{enumerate}
\item Identify type of slice being used
\item Identify formula for the slice
\item Set up and find definite integral using appropriate limits
\end{enumerate}
\newpage
\begin{center}
3 shapes of the slices
\end{center}
\begin{enumerate}


\item Circles (or cylinders)--Area is in contact with the horizontal axis of revolution
\begin{enumerate}
\item Volume of revolution of $f(x)=x^2$ on [0,4] around $x$-axis
\item Volume of revolution of $f(x)=x^3+2$ on [0,3] around $x$-axis
\item Volume of revolution of $f(x)=2x+3$ on [-2,5] around $y=3$
\end{enumerate}
\item Washers--Area is not in contact with horizontal axis of revolution
\begin{enumerate}
\item Volume of revolution of area between $f(x)=x^2$ and $g(x)=8\sqrt{x}$ on [0,4] around $x$-axis
\item Volume of revolution of area between $f(x)=x^2$ and $g(x)=x^3$ on [0,1] around $x$-axis
\item Volume of revolution of area between $f(x)=x^2$ and $g(x)=x^3$ on [0,1] around line $y=4$
\end{enumerate}
\item Cylindrical Shells--Vertical axis of revolution
\begin{enumerate}
\item Volume of revolution of area between $f(x)=x^2+4$ and $g(x)=x^3$ on [0,2] around $y$-axis
\item Volume of revolution of area between $f(x)=x^2$ and $g(x)=x^3$ on [0,1] around $y$-axis
\item Volume of revolution of area between $f(x)=x^2$ and $g(x)=8\sqrt{x}$ on [0,4] around $x=4$
\end{enumerate}
\begin{center}
Arc length
\end{center}
Using the Pythagorean theorem, we can approximate the length of the curve with a Riemann sum.  Therefore, we can find the exact value of this sum, and therefore the length of the arc over $[a,b]$ with the following formula:

$$\mathtt{Arc\,Length}=\int_a^b\,\sqrt{1+(f'(x))^2}\,dx$$
\end{enumerate}
\end{document}