Homework 7 - Volumes of Revolution

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Please enter in all answers as exact answers (for example, if your answer is $3\pi$ , enter in $3\pi$ , not $9.425$ .

1 :

Find the volume $V$ of the solid that is generated when the region bounded by $y = \sec (x), x = \pi /4, x = \pi /3,$ and $y =0$ is revolved around the $x$ axis.

$V = \answer {(\sqrt {3}-1)\pi }$

2 :

Find the volume $V$ of the solid that is generated when the region bounded by $y=x^2+1$ and $y=x+3$ is revolved about the $x$ axis.

$V = \answer {117\pi /5}$

3 :

Find the volume $V$ of the solid obtained by revolving the region enclosed by $x=y^2$ and $x=y+2$ around the $y$ axis.

$V = \answer {72\pi /5}$

4 :

A certain solid has a height of $1$ , and a horizontal cross section taken at a height $x$ above the bottom of the solid is an annulus of inner radius $x^2$ and outer radius $\sqrt {x}$ . Find the volume $V$ of the solid.

$V = \answer {3\pi /10}$

5 :

Find the volume $V$ of the solid generated by revolving the region between $xy = 4$ and $x+y = 5$ around the $x$ axis.

$V = \answer {9\pi }$

6 :

The base of a certain solid is the region enclosed by $y=\sqrt {x},y=0,$ and $x=4$ . Every cross section perpendicular to the $x$ axis is a semi circle with its diameter across the base. Find the volume $V$ of the solid.

$V = \answer {\pi }$

7 :

Find the volume $V$ of the solid generated by revolving the region between $x^2+y^3=4, x=0, x=2, y=0$ about the $y$ axis.

$V = \answer {3\cdot 4^{1/3}\cdot \pi }$

8 :

Find the volume $V$ of the solid generated by revolving the region between the curves $y=1/x, y=0,x=1,x=3$ about the $y$ axis.

$V = \answer {4\pi }$

9 :

The base of a certain solid is the region enclosed between the curves $y = \sin (x), y=0, x = \pi /4, x = 3\pi /4$ , and every cross section perpendicular to the $x$ axis is a square with one side across the base. Find the volume $V$ of the solid.

$V = \answer {(\pi +2)/4}$

10 :

Let $V_x$ and $V_y$ be the volumes of the solids obtained by revolving the region enclosed by $y = 1/x, y=0, x = 1/2$ and $x = b, (b > 1/2)$ about the $x$ and $y$ axes respectively. Is there a value for $b$ for which $V_x=V_y$ ? (If there are multiple values, enter any one of them, if there are no values, enter $0$ ).

$b = \answer {1}$

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Type...	...to get
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