Volumes with Known Cross Sections Worksheet

**Page 1**

1.

What is the volume of a body bounded by the planes $x$ = 2, $x$ = 7 whose area of cross section perpendicular to $x$-axis is inversely proportional to the square of the distance of the section from the origin and the area of the cross section at $x$ = 4 is 13 square units?

a. | 41.6 | ||

b. | 77.29 | ||

c. | 74.29 | ||

d. | 38.6 |

[Given.]

So, A(

[For some constant

A (4) = 13

[Given.]

[Substitute

So, A(

The volume of the solid bounded by

=

= 208(-

=

[Simplify.]

Correct answer : (3)

2.

If $x$ is the distance from the origin to the cross section perpendicular to $x$-axis of a solid, then $x$^{2} represents the area of the cross section. Find the volume of the solid between the planes $x$ = 1, $x$ = 5 .

a. | 3 cubic units | ||

b. | 124 cubic units | ||

c. | $\frac{124}{3}$cubic units | ||

d. | $\frac{3}{124}$cubic units |

The volume of the solid bounded by

V =

=

=

=

[Simplify.]

Correct answer : (3)

3.

A($x$) = $x$^{2} + 6$x$ represents the area of cross section perpendicular to $x$-axis of a solid when $x$ represents the distance of cross section from the origin. What is the volume of the solid bounded by $x$ = 3, $x$ = 6?

a. | 81 cubic units | ||

b. | 144 cubic units | ||

c. | 63 cubic units | ||

d. | 198 cubic units |

The volume of the solid bounded by

= V =

=

[Substitute A(

=

= (

= 144 cubic units

[Simplify.]

Correct answer : (2)

4.

A square based pyramid of height 8 cm is resting on $x$-axis so that its square cross sections are perpendicular to $x$-axis. The vertex of the pyramid is on the plane $x$ = 1 and the base of it is on the plane $x$ = 5. If $x$ is the distance from origin to the cross section, then the area of the cross section is 12$x$^{2}. Find the volume of the pyramid.

a. | 504 | ||

b. | 496 | ||

c. | 372 | ||

d. | 500 |

The volume of the pyramid bounded by

=

[Substitute A(

=

= 4 (125 - 1) = 496

Correct answer : (2)

5.

A solid is lying alongside the interval [0, $\frac{\pi}{4}$] on the $y$-axis. 9sec $y$ tan $y$ is the area of the cross section of the solid perpendicular to the $y$-axis at the point $y$ of [0, $\frac{\pi}{4}$]. What is the volume of the solid?

a. | 9$\sqrt{2}$ - 9 | ||

b. | - 9 - 9$\sqrt{2}$ | ||

c. | 9 - 9$\sqrt{2}$ | ||

d. | 9$\sqrt{2}$ + 9 |

[Given.]

The volume of the solid between

=

[Substitute A(

= 9[sec

= 9 [sec

= 9

Correct answer : (1)

6.

What is the volume of a solid bounded by the planes $x$ = 1, $x$ = 3 whose area of cross section perpendicular to $x$-axis is proportional to ln$x$ where $x$ is the distance of the cross section from the origin, and whose area of cross section is ln 8 square units at $x$ = 2 ?

a. | (9 ln 3) cubic units | ||

b. | (- 9 ln 3 + 6) cubic units | ||

c. | (9 ln 3 + 6) cubic units | ||

d. | (9 ln 3 - 6) cubic units |

A(

[Given.]

A(

[For some constant

A(2) = ln 8

[Given.]

[Substitute

So, A(

[Substitute

The volume of the solid bounded by

=

[Substitute A(

= 3((

= 3((

= 3(3 ln 3 - 2)

= (9 ln 3 - 6) cubic units

[Simplify.]

Correct answer : (4)

7.

The volume of a solid bounded by the planes $x$ = $\frac{\pi}{4}$, and $x$ = $\frac{\pi}{2}$, whose area of cross section perpendicular to the $x$-axis at $x$ is 7sin $x$. Find the volume of the solid.

a. | $\frac{7}{\sqrt{2}}$ cubic units | ||

b. | 2 cubic units | ||

c. | $\sqrt{2}$ cubic units | ||

d. | $\frac{7}{2}$ cubic units |

The volume of the solid between

V =

=

[Substitute A(

= 7 (- cos

=

[Simplify.]

Correct answer : (1)

8.

If $x$ is the distance from the origin to the cross section perpendicular to $x$-axis of a solid, then 12 cos $x$ represents the area of cross section. What is the volume of the solid between the planes $x$ = $\frac{\pi}{3}$ and $x$ = $\frac{\pi}{2}$ ?

a. | 6 - 6$\sqrt{3}$ | ||

b. | 12 - 6$\sqrt{3}$ | ||

c. | 12 + 6$\sqrt{3}$ | ||

d. | 12 - $\sqrt{3}$ |

The volume of the solid between the planes

V =

=

[Substitute A(

= 12(sin

=

= 6(2 -

= 12 - 6

[Simplify.]

Correct answer : (2)

9.

A($x$) = 15tan $x$ is the area of cross section perpendicular to the $x$-axis of a solid, where $x$ is the distance of the cross section from the origin. What is the volume of the solid between the planes $x$ = $\frac{\pi}{6}$ to $x$ = $\frac{\pi}{3}$ ?

a. | $\frac{\mathrm{ln2}}{2}$ | ||

b. | $\frac{\mathrm{15(ln3)}}{2}$ | ||

c. | $\frac{\mathrm{15(ln2)}}{2}$ | ||

d. | $\frac{\mathrm{ln3}}{2}$ |

The volume of the solid between the planes

=

[Substitute A(

= 15(ln sec

= 15(ln sec

= 15(

[Simplify.]

=

Correct answer : (2)

10.

A($x$) = ($\frac{8}{x}$) is the area of cross section perpendicular to the $x$-axis of a solid where $x$ is the distance of the cross section from the origin. Find the volume of the solid between the planes $x$ = 1, $x$ = 17 .

a. | 17(ln 8) cubic units | ||

b. | 8(ln 17) cubic units | ||

c. | ln 17 cubic units | ||

d. | ln 8 cubic units |

The volume of the solid between the planes

=

[Substitute A(

= 8(ln

= 8(ln 17) cubic units

[Simplify.]

Correct answer : (2)