Volume: Shell Method Worksheet

**Page 1**

1.

Find the volume of the solid of revolution formed by revolving the region bounded by $f$($x$) = $x$^{2} and the $x$-axis (0 ≤ $x$ ≤ 2) about $y$-axis. [Use shell method.]

a. | 4$\pi $ cubic units | ||

b. | 32$\pi $ cubic units | ||

c. | 8$\pi $ cubic units | ||

d. | 12$\pi $ cubic units | ||

e. | 16$\pi $ cubic units |

In the shell method, the volume of solid of revolution formed by revolving the region bounded by

Volume of solid of revolution,

= 2

= 2

= 2

Correct answer : (3)

2.

Find the volume of the solid of revolution formed by revolving the region bounded by $f$($y$) = $y$ - $y$^{2} and the $y$-axis (0 ≤ $y$ ≤ 1) about $x$-axis. [Use shell method.]

a. | $\frac{\pi}{6}$ cubic units | ||

b. | $\frac{7\pi}{12}$ cubic units | ||

c. | $\frac{\pi}{12}$ cubic units | ||

d. | $\frac{\pi}{3}$ cubic units | ||

e. | $\frac{14\pi}{12}$ cubic units |

In the shell method, the volume of solid of revolution formed by revolving the region bounded by

Volume of solid of revolution,

= 2

= 2

= 2

= 2

Correct answer : (1)

3.

Find the volume of the solid of revolution formed by revolving the region bounded by $f$($x$) = $e$^{$x$}, $x$-axis, $x$ = 0 and $x$ = 2 about $y$-axis. [Use shell method.]

a. | 2$\pi $ (1 - $e$ ^{2}) cubic units | ||

b. | $\frac{\pi}{2}$ (1 - $e$ ^{2}) cubic units | ||

c. | $\pi $ (1 - $e$ ^{2}) cubic units | ||

d. | $\pi $ (1 + $e$ ^{2}) cubic units | ||

e. | 2$\pi $ (1 + $e$ ^{2}) cubic units |

In shell method, the volume of solid of revolution formed by revolving the region bounded by

Volume of the solid of revolution = 2

= 2

= 2

= 2

= 2

Correct answer : (5)

4.

Find the volume of the solid of revolution formed by revolving the region bounded by $f$($x$) = sin $x$, $x$-axis, $x$ = 0 and $x$ = $\frac{\pi}{2}$ about $y$-axis. [Use shell method.]

a. | 2$\pi $ cubic units | ||

b. | 4$\pi $ cubic units | ||

c. | $\frac{\pi}{2}$ cubic units | ||

d. | $\frac{\pi}{4}$ cubic units | ||

e. | $\pi $ cubic units |

In the shell method, the volume of solid of revolution is formed by revolving the region bounded by

Volume of solid of revolution,

= 2

= 2

Correct answer : (1)

5.

Find the volume of the solid of revolution formed by revolving the region bounded by $f$($y$) = $\sqrt{y}$ and the $y$-axis (0 ≤ $y$ ≤ 2) about $x$-axis. [Use shell method.]

a. | $\frac{16\pi}{5}$ cubic units | ||

b. | $\frac{16\sqrt{2}\pi}{5}$ cubic units | ||

c. | $\frac{8\sqrt{2}\pi}{5}$ cubic units | ||

d. | $\frac{32\pi}{5}$ cubic units | ||

e. | $\frac{8\pi}{5}$ cubic units |

In the shell method, the volume of solid of revolution formed by revolving the region bounded by

Volume of solid of revolution,

= 2

= 2

=

=

Correct answer : (2)

6.

Find the volume of the solid of revolution formed by revolving the region bounded by $f$($x$) = 1 + $\frac{4}{x}$ and the $x$-axis (2 ≤ $x$ ≤ 4) about $y$-axis. [Use shell method.]

a. | 28$\pi $ | ||

b. | 48$\pi $ | ||

c. | 56$\pi $ | ||

d. | 68$\pi $ | ||

e. | 14$\pi $ |

In the shell method, the volume of solid of revolution formed by revolving the region bounded by

Volume of solid of revolution,

= 2

= 2

= 2

= 28

Correct answer : (1)

7.

Find the volume of the solid of revolution formed by revolving the region bounded by $f$($x$) = ln $x$, $x$-axis, $x$ = 1 and $x$ = $e$ about $y$-axis. [Use shell method.]

a. | 2$\pi $ ($e$ ^{2} - 1) cubic units | ||

b. | $\pi $ ($e$ ^{2} - 1) cubic units | ||

c. | 2$\pi $ ($e$ ^{2} + 1) cubic units | ||

d. | $\frac{\pi}{2}({e}^{2}+1)$ cubic units | ||

e. | $\frac{\pi}{2}({e}^{2}-1)$ cubic units |

In the shell method, the volume of solid of revolution formed by revolving to region bounded by

Volume of solid of revolution,

= 2

= 2

= 2

=

Correct answer : (4)

8.

Find the volume of the solid of revolution formed by revolving the region bounded by $f$($y$) = $y$^{2} + $y$ + 1 , $y$-axis, $y$ = 1 and $y$ = 2 about $x$-axis. [Use shell method.]

a. | $\frac{91\pi}{6}$ cubic units | ||

b. | $\frac{41\pi}{3}$ cubic units | ||

c. | $\frac{23\pi}{3}$ cubic units | ||

d. | $\frac{35\pi}{3}$ cubic units | ||

e. | $\frac{33\pi}{3}$ cubic units |

In the shell method, the volume of solid of revolution formed by revolving the region bounded by

Volume of solid of revolution,

= 2

= 2

= 2

Correct answer : (1)

9.

Find the volume of solid formed by revolving the region bounded by $y$ = 2$\sqrt{2x}$ and $y$ = $x$^{2} about $x$-axis. [Use Washer method.]

a. | $\frac{24\pi}{5}$ cubic units | ||

b. | $\frac{56\pi}{5}$ cubic units | ||

c. | $\frac{112\pi}{5}$ cubic units | ||

d. | $\frac{48\pi}{5}$ cubic units | ||

e. | $\frac{32\pi}{5}$ cubic units |

Equations of curves are

These two curves intersect at (0, 0) and (2, 4). So,

In Washer method, the volume of solid formed by revolving the region bounded by

Volume of solid of revolution,

=

=

Correct answer : (4)

10.

Find the volume of solid formed by revolving the region bounded by $y$ = 0 and $y$ = $x$^{2} - 4 about $x$-axis. [Use Washer method.]

a. | $\frac{512\pi}{15}$ cubic units | ||

b. | 47$\pi $ cubic units | ||

c. | 51$\pi $ cubic units | ||

d. | 17$\pi $ cubic units | ||

e. | 85$\pi $ cubic units |

Equations of curves are

These two curves intersect at (- 2, 0) and (2, 0). So,

In Washer method, the volume of solid formed by revolving the region bounded by

Volume of solid of revolution,

= 2

= 2

= 2

Correct answer : (1)