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Volumes with Known Cross Sections Worksheet

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


Solution:

The area of cross section perpendicular to x-axis of the solid = A(x) 1x2 where x is the distance of the cross section from the origin.
[Given.]

So, A(x) = kx2
[For some constant k of proportion.]

A (4) = 13
[Given.]

k16 = 13 k = 208
[Substitute x = 4 in A(x) = kx2.]

So, A(x) = 208x2

The volume of the solid bounded by x = 2, x = 7 is V = 27 A(x)dx

= 27 208x2dx

= 208(- 1x)27

= 1040 / 14 = 74.29
[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 x2 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.
124 3cubic units
d.
3 124cubic units


Solution:

Area of cross section perpendicular to x-axis of the solid = A(x) = x2 where x is the distance of the cross section from the origin.

The volume of the solid bounded by x = 1, x = 5 is

V = 15 A(x)dx

= 15 x2dx

= 1 / 3(x3)15

= 124 / 3 cubic units
[Simplify.]


Correct answer : (3)
 3.  
A(x) = x2 + 6x 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


Solution:

Area of cross section perpendicular to x-axis of the solid = A(x) = x2 + 6x where x is the distance of the cross section from the origin.

The volume of the solid bounded by x = 3, x = 6

= V = 36 A(x)dx

= 36 (x2 + 6x)dx
[Substitute A(x) = x2 + 6x.]

= 1 / 3(x3)36 + 3(x2)36

= (1 / 3)(216 - 27) + 3(36 - 9)

= 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 12x2. Find the volume of the pyramid.
a.
504
b.
496
c.
372
d.
500


Solution:

Area of cross section perpendicular to x-axis of the pyramid = A(x) = 12x2 where x is the distance of the cross section from the origin.

The volume of the pyramid bounded by x = 1, x = 5 is V = 1 5 A(x)dx

= 1 5(12x2) dx
[Substitute A(x) = 12x2.]

= 12 / 3(x3)1 5

= 4 (125 - 1) = 496


Correct answer : (2)
 5.  
A solid is lying alongside the interval [0, π4] 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, π4]. What is the volume of the solid?
a.
92 - 9
b.
- 9 - 92
c.
9 - 92
d.
92 + 9


Solution:

The area of cross section perpendicular to the y-axis of the solid is A(y) = 9sec y tan y where y [0, π4]
[Given.]

The volume of the solid between y = 0, y = π4 is V = 0π/4 A(y) dy

= 0π/4 9 sec y tan y dy
[Substitute A(y) = 9 sec y tan y.]

= 9[sec y]0π/4

= 9 [sec π4 - sec 0]

= 92 - 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 lnx 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


Solution:

The area of cross section perpendicular to x - axis of the solid is proportional to ln x.

A(x) ln x
[Given.]

A(x) = k ln x
[For some constant k of proportion.]

A(2) = ln 8
[Given.]

k ln 2 = 3 ln 2 k = 3
[Substitute x = 2 in A(x) = k ln x.]

So, A(x) = 3 ln x
[Substitute k = 3 in A(x) = k ln x.]

The volume of the solid bounded by x = 1, x = 3 is V = 1 3 A(x)dx

= 1 3 3ln x dx
[Substitute A(x) = 3 ln x.]

= 3((xln x)13 - 13dx)

= 3((x ln x)13 - (x)13)

= 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 = π4, and x = π2, whose area of cross section perpendicular to the x-axis at x is 7sin x. Find the volume of the solid.
a.
72 cubic units
b.
2 cubic units
c.
2 cubic units
d.
7 2 cubic units


Solution:

Area of cross section of the solid perpendicular to x - axis = A(x) = 7 sin x where x is the distance of the cross section from the origin.

The volume of the solid between x = π4, x = π2 is

V = π/4π/2 A(x) dx

= π/4π/2 7sin x dx
[Substitute A(x) = 7sin x.]

= 7 (- cos x)π/4π/2

= 72 cubic units.
[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 = π3 and x = π2 ?
a.
6 - 63
b.
12 - 63
c.
12 + 63
d.
12 - 3


Solution:

The area of cross section perpendicular to x - axis whose distance from origin is x = A(x) = 12 cos x.

The volume of the solid between the planes x = π3, x = π2 is

V = π/3π/2 A(x) dx

= π/3π/2 12 cos x dx
[Substitute A(x) = 12 cos x.]

= 12(sin x)π/3π/2

= 12(2 -3)2

= 6(2 - 3)

= 12 - 63
[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 = π6 to x = π3 ?
a.
ln2 2
b.
15(ln3) 2
c.
15(ln2) 2
d.
ln3 2


Solution:

The area of cross section of the solid perpendicular to the x-axis whose distance from the origin is x = A(x) = 15tan x

The volume of the solid between the planes x = π6, x = π3 is V = π/6π/3 A(x)dx

= π/6π/3 15tan x dx
[Substitute A(x) = 15tan x.]

= 15(ln sec x)π/6π/3

= 15(ln sec π3 - ln sec π6)

= 15(1 / 2) ln 3
[Simplify.]

= 15(ln3) / 2


Correct answer : (2)
 10.  
A(x) = (8x) 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


Solution:

The area of cross section perpendicular to the x-axis of the solid = A(x) = 8x where x is the distance from the origin to the cross section.

The volume of the solid between the planes x = 1, x = 17 is V = 117 A(x) dx

= 117 8x dx
[Substitute A(x) = 8x.]

= 8(ln x)117

= 8(ln 17) cubic units
[Simplify.]


Correct answer : (2)

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