Volumes of Solids of Revolution

CHAPTER 16

In an earlier chapter we saw how integration can be used to evaluate areas of certain regions. Similarly integration can be used to find volumes of solids formed by rotating a region about a given straight line. The method used is analogous to the technique of using thin strips to find areas.

For example consider the region bounded by , the -axis and the
-axis:

When is rotated 360 ° about the -axis a cone is formed as shown below:

To find the volume of the cone we could simply use the formula
i.e. but instead we will use this as an introductory example of the “strip” method for finding volumes.

Consider a thin horizontal strip as shown below whose length is and whose width is .

Remember that refers to the co-ordinate of a point () on the graph and therefore does represent the length of the mini strip.

When the thin strip is rotated 360 ° about the -axis it forms a thin disc whose volume is .

Note that the cone is composed of an infinite number of thin horizontal strips whose sum will yield the cone. It follows that the volume of the cone will be .

Hence the volume of the cone

as previously shown.

When evaluating the volume of a solid of revolution it is important to concentrate upon what happens to the thin strip when rotated rather than the whole solid itself.

Example

Disc Method

Question:The region bounded by , , , and the axis, is rotated
360 ° about the -axis. Find the volume of the figure so formed.

Answer:

Consider a thin vertical strip width , height . When it is rotated about the -axis it forms a disc whose volume is .
The volume of the solid

(approx.)

Example

Washer Method

Question:The region bounded by and is rotated 360 ° about the
-axis. Find the volume of the solid so formed.

Answer:

intersects at (2,8) and (0,0). We wish to find the volume of the “crescent” shaped region rotated around the -axis.

Consider a thin vertical strip width .

When it is rotated around the -axis it becomes a “washer”

whose volume is where ,
refer to the radii of the large and small circle respectively.

Note that and .

Volume of the washer

Volume of solid of revolution

Example

In the previous example, if the region had been rotated about the -axis then its volume could have been found as follows.

Consider a thin horizontal strip which, when rotated about the -axis, would again form a “washer” whose volume is where , are the large and small radii respectively.

is the co-ordinate of a point on and is the co-ordinate of a point on . i.e. and .

The volume of the washer

The volume of the solid of revolution

Frequently it is possible to consider either horizontal or vertical strips but in certain instances it is either impractical or impossible to use one of them.

Example

Question:Consider the region bounded by and . Find the volume of the solid formed when this region is rotated about the
-axis.

Answer:

Clearly it is impractical to use horizontal strips for values of greater than 2 and hence a vertical strip is used. The thin vertical strip, when rotated about the -axis produces a washer whose volume is .

The volume of the solid of revolution:

(approx.)

Example

Shell Method

Question:Consider the solid formed when the region in the last example is rotated about the -axis. Find the volume of the solid so formed.

Answer:As explained it is impractical to use a horizontal strip and hence the thin vertical strip, when rotated 360 ° about the -axis produces a shape known as a shell whose dimensions are shown in the figure below. The word shell is used in its military, rather than marine, sense.

To evaluate the volume of the shell so formed, imagine cutting open the shell to produce a lamina as shown below

whose dimensions are as shown.

The volume of the lamina is .

The volume of the solid of revolution formed is

Example

Question:The region bounded by the -axis and is rotated around the vertical line . Find the volume of the solid so formed.

Answer:

When rotated the vertical thin strip becomes a solid as shown below.

The volume of this thin shell considered as a lamina when cut open is

Total volume of solid of revolution

Worksheet 1

1.Find the volume of the solid of revolution formed by rotating the
ellipse a) about the -axis.b) about the -axis.

2.Find the volume of the solid generated by rotating about the -axis, the area formed by , the -axis, and .

3.Find the volume of the solid generated by rotating about the line , the area bounded by , the -axis, and .

4. is the origin, is (2,0), is (2,2), and is (0,2).
a) Find the volume of the solid of revolution formed by rotating triangle
about the -axis.
b) Find the volume of the solid of revolution formed by rotating triangle
about the -axis.
c) Since clearly area of = area of , why are the volumes
generated different?

5.Find the volume of the solid generated by rotating the region bounded by and a) about the -axis and b) about the line .

6.The graph of is drawn below.

Find the volume of the solid generated by rotating the loop around the -axis.

7.The area bounded by the curve , the -axis, , and is rotated about the -axis. Find the volume of the solid figure so formed.

Answers to Worksheet 1

1.a) b) 2. 3. 4. a) b)

5.a) b) 6. 7.

Worksheet 2 – Calculators Permitted

The region is bounded above by the graph of , on the left by , on the right by , and below by . The volume of the solid of revolution formed when is revolved about the line is nearest in value to:
(A) 6.8(B) 7.0(C) 7.2(D) 7.4(E) 7.6

The region in the first quadrant is bounded by and . If is revolved about the -axis, the volume of the solid formed is nearest in value to:
(A) 8.16(B) 8.26(C) 8.36(D) 8.38(E) 8.42

Let be the region in the first quadrant bounded by the -axis, the line and the graph of . What is the volume of the solid generated by rotating about the -axis?
(A) 2.67(B) 2.70(C) 2.73(D) 2.76(E) 2.79

Let be the region in the first quadrant boundedby the -axis and the curve . The volume produced when is revolved about the -axis is:
(A) (B) (C) (D) (E)

Let be the region in the first quadrant bounded above by the graph of and below by the graph of . What is the volume of the solid generated when is rotated about the -axis?
(A) 1.21(B) 2.68(C) 4.17(D) 6.66(E) 7.15
Let be the region enclosed by the graphs of , , and the line . The volume of the solid generated when is revolved about the
-axis is nearest to
(A) 33.09(B) 33.11(C) 33.13(D) 33.15(E) 33.17
Let be the region in the first quadrant enclosed by the graphs of and the lines and . What is the volume of the solid generated when is rotated about the -axis?
(A) 15.9(B) 18.7(C) 40.1(D) 50.6(E) 64.9
Which definite integral represents the volume of a sphere with radius 2?
(A) (B) (C)
(D)(E)

Answers to Worksheet 2

A 2. D 3. B 4. A 5. E 6. B7. C 8. C

Volumes of Solids with Known Cross-Section

It is important to remember that integration is the process of adding together an infinite number of small “things”. For example suppose we had a closed vessel whose cross-section area, parallel to the base, was square metres where represented the distance in metres of the cross-section region from the base.

The base area, when , would be 36 square metres and the height of the vessel would be 6 metres because, when the cross-section area equals zero. The vessel would resemble a “beehive” as shown.

If we wished to find the volume of the “beehive” then we would simply evaluate since represents the volume of a thin horizontal strip and the integration process adds them all up.

Volume cubic metres.

Similarly consider the following question taken from an Advanced Placement Calculus AB examination.

Let be the region in the first quadrant enclosed by the graphs of , , and the -axis, as shown in the figure above.
a) Find the area of the region .
b) Find the volume of the solid generated when the region is revolved about the
-axis.
c) The region is the base of a solid. For this solid, each cross section
perpendicular to the -axis is a square. Find the volume of this solid.

Answer: a) We need to find first the point of intersection of and
. By calculator, is (0.94194408, 0.41178305).

By considering a thin vertical strip it is clear that the area of region (approx.)

b) Similarly by considering a thin vertical strip, when rotated around
the -axis, it becomes a washer whose volume is
.

The volume of the solid of revolution so formed is

(approx.)

c) The volume of eachsquare cross-section is .

and hence the total volume of the solid is

Example

Question:Shown below is the base of a solid represented by . The solid is formed by having cross-sections in planes perpendicular to the -axis between and and are isosceles right-angled triangles with one side in the base. Find the volume of the solid.

Answer:Each cross section has width and area .

each cross section has volume .

The total volume is hence

An interesting theorem concerning rotations of regions is Pappus’ Theorem which states that the volume of a solid formed by rotating a region about a line, not intersecting the region, equals the area of the region times the distance travelled by the centre of gravity of the region in the rotation.

For example, consider the volume of the solid formed by rotating the circle about the line .

The volume of the torus (doughnut) so formed

= (Area of circle) times (circumference of circle formed when (0,0) rotates
360 ° around )

= times

= 3947.8 (approx.)

Worksheet 3

1.The area bounded by the -axis, the -axis, and is rotated about the -axis. Find the volume of the solid so formed.

2.A hemispherical bowl of radius 5 cm contains water to a depth of 3 cm. Find the volume of the water.

3.The area in the first quadrant bounded by , , and the -axis is rotated about the -axis. Find the volume of the solid generated.

4.Prove that the volume of a sphere is considering the rotation of the circle about the -axis.

5.A container is such that its cross-section area is cm2 where is the distance in cm from the base.
a) What is the height of the container?
b) What is its volume?

6.The area of a cross-section of a vase at a distance cm below the top is cm2. Find the depth of the water when the vase is half full.

7.Find the volume of the solid generated by rotating the area bounded by and
a) about the -axis.
b) about the -axis.

8.Find the volume generated when the region enclosed by the lines , , and between (1,1) and (3, ) is rotated about the -axis.

9.A closed vessel tapers to point and at its ends and is such that its cross-section area cut by a plane perpendicular to , cm from , is
cm2. Find the volume of the vessel.

10. The area bounded by the -axis, , and is rotated about the
-axis. Find the volume of the solid generated.

11. Find the volume formed by rotating the area bounded , the -axis and ,
a) about the -axis.
b) about the line .

12. A solid is 12 inches high. The cross-section of the solid at height above its base has area square inches. Find the volume of the solid.

13. A solid extends from to . The cross-section of the solid in the plane perpendicular to the -axis is a square of side . Find the volume of the solid.

14. A solid is 6 ft high. Its horizontal cross-section at height ft above the base is a rectangle with length ft and width ft. Find the volume of the solid.

15. A solid extends along the axis from to . Its cross-section at any point is an equilateral triangle with edge . Find the volume of the solid.

Answers to Worksheet 3

48.592. 3. 4. ----5. a) 2b)

0.6957. a) b) 8. 9.

11. a) b) 12. 21613.

13215.

Worksheet 4

The finite area in the first quadrant bounded by the curves , and the line is rotated once about the -axis. Find the volume of the solid formed.
Find the area enclosed by , and the -axis.
Find the volume of the solid formed by rotating the circle about the -axis. (Pappus’ Theorem required)
The area bounded by the -axis, and is rotated about the line . Find the volume of the solid generated.
Find the volume of the solid generated by rotating about the -axis, the region, in the first quadrant, bounded by and .
Find the volume of the solid formed by rotating the area bounded by , the -axis and about the line .
Find the volume of the solid formed by rotating about its major axis.
The region in the first quadrant bounded by is rotated about the Find the volume of the solid formed.

A hole of radius is drilled through a solid sphere of radius , with one edge of the hole passing through the centre of the sphere. The volume of the material removed is where is integer. Find the value of .
Find the volume of the solid formed by rotating around the -axis. (This means the finite area above the -axis between and ).
Find the volume of the solid formed by rotating the area enclosed by , and the -axis around the -axis.

Answers to Worksheet 4

2. 5.96983. 4.

6. 7. 8.

8 10. 3.586411. 28.15

Worksheet 5

Let be the region in the first quadrant bounded by the graphs of and the line .
a) Find the area of in terms of .
b) Find the volume of the solid formed when is rotated 360 ° about the -
axis.
c) Find the volume in part b) as .
Let be the shaded region in the first quadrant enclosed by the -axis and the graphs of and as shown in the figure below.

a) Find the area of .
b) Find the volume of the solid generated when is revolved about the
-axis.
c) Find the volume of the solid whose base is and whose cross sections

perpendicular to the -axis are squares.

Let be the region in the first quadrant under the graph of for .
a) Find the area of .
b) If the line divides the region into two regions of equal area, what
is the value of ?
c) Find the volume of the solid whose base is the region and whose
cross-sections cut by planes perpendicular to the -axis are squares.
Let be the region enclosed by the graphs of and .
a) Find the area of .
b) The base of a solid is the region . Each cross-section of the solid
perpendicular to the -axis is an equilateral triangle. Find the volume of
the solid.
Find the volumes of the solids described:
a) The solid lies between planes perpendicular to the -axis at and
. The cross-sections perpendicular to the -axis are circular discs
with diameters running from the -axis to the parabola .
b) The solid lies between planes perpendicular to the -axis at and
. The cross-sections perpendicular to the -axis between these
planes are squares whose diagonals run from the semicircle
to the semicircle .

Answers to Worksheet 5

a) b) c)
a) 1.764b) 30.46c) 3.671
a) 2b) c) 0.811 (approx.)
a) 1.168 b) 0.3967 (approx.)
a) b)