VOLUMES OF SOLIDS OF REVOLUTION

• Questions involving the area of a region between curves, and the volume of a solid formed when this region is rotated about a horizontal or vertical line, appear regularly on both the AP Calculus AB and BC exams.

• Students have difficulty when the solid is formed by use a line of rotation other than the x- or y-axis.

• These types of volume are part of the type of volume problems students must solve on the AP test.

• Students should find the volume of a solid with a known cross section.

• The Shell method is not part of the AB or the BC course of study anymore.

• The four examples in the Curriculum Module use the disk method or the washer method.

Example 1 Line of Rotation Below the Region to be Rotated

• Picture the solid (with a hole) generated when the region bounded by and are revolved about the line y = -2.

• First find the described region• Then create the reflection

over the line y=-2

2y x xy e -1 1

-6

-5

-4

-3

-2

-1

1

Example 1

• Think about each of the lines spinning and creating the solid.

• Draw one representative disk.

• Draw in the radius.

Example 1• Find the radius of the

larger circle, its area and the volume of the disk. 2 outsider y

22 outsideArea y

2

2

2

2 2

outsideVolume y x

x x

• Sum up these cylinders to find the total volume

21

2 2n

k

Volume x x

The larger the number of disks and the thinner each disk, the smoother the stack of disks will be. To obtain a perfectly smooth solid, we let n approach infinity and Δx approach 0.

• The points of intersection can be found using the calculator.

• Store these in the graphing calculator

(A=-1.980974,B=0.13793483)(C=0.44754216,D=1.5644623)

• Write an integral to find the volume of the solid.

22 2 71.9833363C

A

Volume x dx

Example 1• Find the radius of the

smaller circle, its area and the volume of the disk. 2 insider y

22 insideArea y

2

2

2

2

inside

x

Volume y x

e x

• Sum up these cylinders to find the total volume

21

2n

x

k

Volume e x

The larger the number of disks and the thinner each disk, the smoother the stack of disks will be. To obtain a perfectly smooth solid, we let n approach infinity and Δx approach 0.

• Using the points of intersection write a second integral for the inside volume.

(A=-1.980974,B=0.13793483)(C=0.44754216,D=1.5644623)

22 52.258610C

x

A

Volume e dx

Example 1• The final volume will

be the difference between the two volumes.

2 22 2 2 19.724 19.725

C Cx

A A

x dx e dx or

Example 2 Line of Rotation Above the Region to be Rotated

• Rotate the same region about y = 2

• Notice that 2y x xy e

2

2outside

outside

y radius

r y

Example 2 Line of Rotation Above the Region to be Rotated

• The area of the larger circle is

2

2

2

2

outside

x

y

e

22 xVolume e x

Example 2 Line of Rotation Above the Region to be Rotated

• The sum of the volumes is

2

1

2

2

2

16.406065

nx

k

Cx

A

Volume e x

e dx

Example 2 Line of Rotation Above the Region to be Rotated

• The area of the smaller circle is

2

2

2

2 2

insidey

x

2

2 2Volume x x

Example 2 Line of Rotation Above the Region to be Rotated

• The sum of the volumes is

.

2

1

2

2 2

2 2

7 870360

Vn

k

C

A

Volume x x

x dx

Example 2 Line of Rotation Above the Region to be Rotated

• The volume of the solid is the difference between the two volumes

222 2 2

8.535 8.536

C C

A A

Volume

e dx x dx

or

Example 3 Line of Rotation to the Left of the Region to be Rotated

• Line of Rotation: x = -3• Use the same two

functions• Create the reflection• Draw the two disks and

mark the radius

Example 3 Line of Rotation to the Left of the Region to be Rotated

• The radius will be an x-distance so we will have to write the radius as a function of y.

2y x xy e

2

2

2

2

y x

x y

ln

xy e

y x

Example 3 Line of Rotation to the Left of the Region to be Rotated

The radius of the larger disk is 3 + the distance from the y-axis or 3 + (ln y) Area of the larger circle is

23 lny

Example 3 Line of Rotation to the Left of the Region to be Rotated

• Volume of each disk:

23 lnVolume y y

23 lnD

BVolume y dy

Example 3 Line of Rotation to the Left of the Region to be Rotated

• The radius of the smaller disk is

• 3+ the distance from the y-axis or 3 + (y2 – 2)

• Area of the larger circle is

223 ( 2)y

Example 3 Line of Rotation to the Left of the Region to be Rotated

• Volume of each disk:

223 ( 2)Volume y y

223 ( 2)D

BVolume y dy

Example 3 Line of Rotation to the Left of the Region to be Rotated

• Difference in the volume is

22 23 (ln ) 3 ( 2)

15.538 15.539

D D

B By dy y dy

or

Example 4 Line of Rotation to the Right of the Region to Be Rotated

• Line of Rotation: x = 1

• Create the region, reflect the region and draw the disks and the radius

Example 4 Line of Rotation to the Right of the Region to Be Rotated

• Notice the larger radius is 1 + the distance from the y-axis to the outside curve.

• The distance is from the y-axis is negative so the radius is

21 ( 2)y

Example 4 Line of Rotation to the Right of the Region to Be Rotated

• Area of Larger disk:

• The volume of the disk is

221 ( 2)y

221 2 y x

Example 4 Line of Rotation to the Right of the Region to Be Rotated

• Volume of all the disks are

22

1

22

1 ( 2)

1 ( 2)

n

k

D

B

y y

y dy

V

Example 4 Line of Rotation to the Right of the Region to Be Rotated

• Area of smaller disk:

• The volume of the disk is

21 (ln )y

21 lny y V

Example 4 Line of Rotation to the Right of the Region to Be Rotated

• Volume of all the disks are

2

1

2

1 ln

1 ln

n

k

D

B

y y

y dy

V

Example 4 Line of Rotation to the Right of the Region to Be Rotated

• Find the difference in the volumes

2 221 ( 2) 1 ln 12.72067D D

B By dy y dy