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\title{Math 1b. Volumes}

\date{Spring 2006}

\begin{document}

\maketitle





\begin{enumerate}

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

\vfill

\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.

\vfill

\clearpage

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


\vfill



\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 $S$?

\vfill


\end{enumerate}



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