Sketch the solid.

Graph

Graph Description

A surface of revolution is generated by a curve rotating about the $\backslash ($ x $\backslash )-$axis. The surface starts at $\backslash ($ x$=0\backslash ),$ with radius $0,$ goes in the positive $\backslash ($ x $\backslash )$ direction, becomes wider, goes through $\backslash ($ x$=0\mathrm{.}4\backslash ),$ with radius $3,$ becomes narrower, and ends at $\backslash ($ x$=0\mathrm{.}6\backslash ),$ with radius $0.$ A vertical line segment at $\backslash ($ x$=0\mathrm{.}4\backslash )$ is rotated about the horizontal line $\backslash ($ y$=0\backslash )$ forming a circle inside the volume of revolution.

Graph

Graph Description

A surface of revolution is generated by a curve rotating about the $\backslash ($ x $\backslash )-$axis. The surface starts at $\backslash ($ x$=0\backslash ),$ with radius $0,$ goes in the positive $\backslash ($ x $\backslash )$ direction, becomes wider, goes through $\backslash ($ x$=0\mathrm{.}7\backslash ),$ with radius $9,$ becomes narrower, and ends at $\backslash ($ x$=1\backslash ),$ with radius $0.$ A vertical line segment at $\backslash ($ x$=0\mathrm{.}7\backslash )$ is rotated about the horizontal line $\backslash ($ y$=0\backslash )$ forming a circle inside the volume of revolution. A surface of revolution is generated by a curve rotating about the $\backslash ($ y $\backslash )-$axis. The surface starts at $\backslash ($ y$=0\backslash ),$ with radius $0,$ goes in the positive $\backslash ($ y $\backslash )$ direction, becomes wider, and ends at $\backslash ($ y$=0\backslash ),$ with radius $1.$ A vertical line segment at $\backslash ($ x$=0\mathrm{.}7\backslash )$ is rotated about the vertical line $\backslash ($ x$=0\backslash )$ forming a cylinder inside the volume of revolution.

A surface of revolution is generated by a curve rotating about the $\backslash ($ y $\backslash )-$axis. The surface starts at $\backslash ($ y$=0\backslash ),$ with radius $0,$ goes in the positive $\backslash ($ y $\backslash )$ direction, becomes wider, and ends at $\backslash ($ y$=0\backslash ),$ with radius $0\mathrm{.}6.$ A vertical line segment at $\backslash ($ x$=0\mathrm{.}4\backslash )$ is rotated about the vertical line $\backslash ($ x$=0\backslash )$ forming a cylinder inside the volume of revolution.

What are the circumference $\backslash ($ c $\backslash )$ and height $\backslash ($ h $\backslash )$ of a typical cylindrical shell?

$\backslash [$

c$($x$)=$

$\backslash ]$

$\backslash [$

h$($x$)=$

$\backslash ]$

Use the method of cylindrical shells to find the volume $\backslash ($ V $\backslash )$ of $\backslash ($ S $\backslash ).$

Do you think this method is preferable to using washers? Explain your reasoning.