Module $3$

$1.(5$ points$)$ A $6-$inch$-$tall plastic cup is shaped like a surface obtained by rotating a line segment in the first quadrant about the $\backslash ($ x $\backslash )-$axis. Given that the radius of the base of the cup is $1$ inch, the radius of the top of the cup is $1\mathrm{.}5$ inches, and the cup is filled to the brim with water, use integration to find the volume of water in the cup. Make a sketch showing the $\backslash ($ x $\backslash )-$ and $\backslash ($ y $\backslash )-$axes and the function you use.

$2.(5$ points$)$ A torus $($donut$)$ is formed by revolving the region bounded by the circle $\backslash (($x$-2)^\{2\}+\backslash )\backslash ($ y$^\{2\}=1\backslash )$ about the $\backslash ($ y $\backslash )-$axis. Make a sketch and use the disk method to write an integral or sum of integrals that would give the volume of the torus. You don't need to evaluate the integral$($s$).$

$3.(5$ points$)$ Use the general slicing method to find the volume of the solid whose base is the region bounded by the curve $\backslash ($ y$=\backslash$sqrt$\{\backslash$cos x$\}\backslash )$ and the $\backslash ($ x $\backslash )-$axis on $\backslash (\backslash$left$[-\backslash$frac$\{\backslash$pi$\}\{2\},\backslash$frac$\{\backslash$pi$\}\{2\}\backslash$right$]\backslash ),$ and whose cross sections through the solid perpendicular to the $\backslash ($ x $\backslash )-$axix are isosceles right triangles with a horizontal leg in the $\backslash ($ x y $\backslash )-$plane and a vertical leg above the $\backslash ($ x $\backslash )-$axis. $($See the figure below.$)$

$4.(5$ points$)$ Let $\backslash ($ R $\backslash )$ be the region bounded by the curves $\backslash ($ y$=$x $\backslash )$ and $\backslash ($ y$=1+\backslash$frac$\{$x$\}\{2\}\backslash ).$ Make a sketch and find the volume of the solid generated if $\backslash ($ R $\backslash )$ is revolved about the line $\backslash ($ y$=3\backslash ).$