Volumes of Revolution ('discs' and 'shells') Worksheet

**Page 1**

1.

What is the volume of the solid generated by rotating about the $x$ - axis the plane region bounded by $y$ = cos $x$ and $y$ = 0 over [0, $\frac{\pi}{2}$]?

a. | $\frac{{\pi}^{2}}{6}$ cubic units | ||

b. | $\frac{{\pi}^{2}}{2}$ cubic units | ||

c. | $\frac{{\pi}^{2}}{4}$ cubic units | ||

d. | $\frac{{\pi}^{2}}{8}$ cubic units |

The plane region R bounded by the curves

The volume of the solid generated by rotating the plane region between

= V =

[Disc method.]

=

[Substitute

=

[Use cos

=

=

=

= (

=

Correct answer : (3)

2.

Find the volume of the solid formed by rotating about the $x$ - axis the plane region bounded by $y$ = $e$^{$x$}, $y$ = 0, $x$ = 0, and $x$ = 1.

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

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

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

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

The plane region bounded by the curves

The volume of the solid generated by rotating the shaded plane region which is in between

= V =

[Disc method.]

=

[Substitute

=

=

=

= (

= (

Correct answer : (3)

3.

Find the approximate volume generated by rotating about the $x$ - axis the plane region bounded by $y$ = 4 - $x$^{2} and $y$ = 0.

a. | 33$\pi $ cubic units | ||

b. | 30$\pi $ cubic units | ||

c. | 34$\pi $ cubic units | ||

d. | 32$\pi $ cubic units |

The curve

[Solve 4 -

The curve

[Put

The plane region bounded by

The volume of the solid generated by rotating the shaded plane region which is in between

= V =

[Disc method.]

=

[Substitute

=

=

=

=

=

= 34

Correct answer : (3)

4.

What is the volume of the solid generated by rotating about the $y$ - axis the plane region bounded by $x$ = $e$^{y}, $x$ = 0, $y$ = 1, and $y$ = 2?

a. | ($\frac{\pi}{2}$) $e$ ^{2}($e$^{2} + 1) | ||

b. | ($\frac{\pi}{2}$)$e$ ^{4} | ||

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

d. | ($\frac{\pi}{2}$) $e$ ^{2} |

The plane region bounded by

The volume of the solid generated by rotating the shaded plane region between

= V =

[Disc Method.]

=

[Substitute

=

=

= (

=

Correct answer : (3)

5.

Find the volume of the solid generated by rotating about the $x$ - axis the region bounded by $x$ = $e$^{$y$}, $x$ = 0, $y$ = 0, and $y$ = 1.

a. | 3$\pi $ cubic untis | ||

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

c. | 2$\pi $ cubic untis | ||

d. | 5$\pi $ cubic untis |

The plane region bounded by

The volume of the solid generated by rotating the shaded region between

= V =

[Shell method.]

= 2

[Substitute

= 2

[Use integration by parts.]

= 2

= 2

Correct answer : (3)

6.

What is the volume of the solid generated by rotating about the $x$ - axis the region bounded by $x$ = ln $y$, $x$ = 0, $y$ = 1, and $y$ = 2?

a. | 2$\pi $ln ^{2} | ||

b. | 2$\pi $(ln 2 - $\frac{3}{4}$) | ||

c. | 2$\pi $ln 4 | ||

d. | 2$\pi $(ln 4 - $\frac{3}{4}$) |

The plane region bounded by

The volume of the solid generated by rotating the shaded region between

V =

[Shell method.]

= 2

[Substitute

= 2

[Use integration by parts.]

= 2

= 2

= 2

Correct answer : (4)