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\begin{document} \begin{frame} The region $R$ is bounded by the curves $$ y=x^3 \hskip 20pt y=8 \hskip 20pt x=0 $$ Sketch $R$. For the following rotational axes, {\bf set-up} two integrals for the volume of the solid generated by revolving $R$ about the indicated axis, one representing the washer method and one the cylindrical shells method. \begin{itemize} \item[\bf (a)] $x$-axis. \item[\bf (b)] $y$-axis. \item[\bf (c)] $y=5$. \item[\bf (d)] $x=-2$. \end{itemize} \end{frame} \begin{frame} The region $R$ is bounded by the curves $$ y=1+\sin(x) \hskip 20pt y=1 \hskip 20pt x=0 \hskip 20pt x=2 $$ Sketch $R$. For the following rotational axes, {\bf set-up} two integrals for the volume of the solid generated by revolving $R$ about the indicated axis, one representing the washer method and one the cylindrical shells method. \begin{itemize} \item[\bf (a)] $x$-axis. \item[\bf (b)] $y$-axis. \item[\bf (c)] $y=-1$. \end{itemize} \end{frame} \begin{frame} The triangular region with vertices $(0,2)$, $(1,0)$, and $(0,1)$ is rotated about the line $x=4$. Find the volume of the solid generated by this rotation. \end{frame} \begin{frame} Let $B$ be the region bounded by the graphs of $x=y^2$ and $x=9$. Sketch $B$. For each part below, find the volume of the solid that has $B$ as its base if every cross section by a plane perpendicular to the $x$-axis is \begin{itemize} \item[\bf (a)] a square. \item[\bf (b)] a semicircle with diameter lying on $B$. \item[\bf (c)] an equilateral triangle. \end{itemize} \end{frame} \begin{frame} Find the volume of a wedge cut out of a cylinder of radius $r$ if the angle between the top and bottom of the wedge is $\frac{\pi}{6}$. \end{frame} \end{document}