7.3.1 Volume by Disks and Washers

I. Solids of Revolution

A.) Def- If a region in the plane is revolved about a line in the plane, the resulting solid is called a SOLID OF REVOLUTION and the line is called the AXIS OF REVOLUTION.

B.) Ex. - Find the volume of the solid of revolution generated by rotating the area bounded by

y = 2x, x = 3, and y = 0 about the x-axis.

What does this solid of revolution look like?

6

4

2

-2

2 4 6

Now, take a slice of the volume by passing two parallel planes perpendicular to the axis of revolution.

6

4

2

-2

2 4 6

The slice is a circular disk obtained by rotating our representative rectangle about the x-axis.

6

4

2

-2

2 4 6

r

Height or dx

An approximation of the VOLUME of the SOR

would be .

The actual volume of the SOR can be found by

2

1

2 n

kk

V x x

3

2 2

01 0

lim 2 = 2n

kx

k

V x x x dx

2r

h

II. Examples Using DisksA.) Ex. 1- Find the volume of the solid of revolution

generated by rotating the area bounded by about the x-axis.225 and 0y x y

4

2

-5 5

2, 25x x

,0x

4

2

-5 5

2, 25x x

,0x

5 2

2

5

= 25V x dx

5

2

5

= 25V x dx

5

3

5

1 = 25

3V x x

3500 = units

3V

B.) Ex. 2 - Find the volume of the solid of revolution generated by rotating the area bounded by , x = 0, and y = 4 about the y-axis.

2y x

,y y 0, y

4 2

0

=V y dy4

0

=V ydy4

2

0

1 =

2V y

3 =8 unitsV

III. Disks – General Case:

A.) Around the x-axis:

B.) Around the y-axis:

2( )

b

a

V f x dx

2( )

d

c

V g y dy

IV. Washers

ALL RADII MEASURED FROM THE AXIS OF

REVOLUTION. 2 2

2 1V r h r h

r1

r2

2 2

2 1V r r h

2 2

2 1

b

a

V r r dr

V. Examples Using WashersA.) Ex. 3- Find the volume of the SOR generated by

rotating the area bdd by about the x-axis.

, 1, and 0y x y x

1

,x x

,1x

1 22

0

= 1V x dx

1

0

= 1V x dx 1

2

0

1 =

2V x x

3 = units

2V

2

1

-1

1 2

B.) Ex. 4- Find the volume of the SOR generated by rotating the area bdd by about the y-axis.

, 1, and 0y x x y

1, y 2 ,y y

1

22 2

0

= 1V y dy

1

4

0

= 1V y dy 1

5

0

1 =

5V y y

34 = units

5V

VI. Washers– General Case:

A.) Around the x-axis:

B.) Around the y-axis:

2 2( ) ( )

b

a

V f x g x dx

2 2( ) ( )

d

c

V f y g y dy

VII. Revolving About a Line Other Than the x and y-axes.1.) Find the volume of the SOR generated by rotating the

area bdd by about the line y=5. 2 and 2y x y x 5

4

3

2

1

-1

1 2

2,x x

, 2x x

2

2 22

0

= 5 5 2V x x dx

2

2 4

0

= 20 14V x x x dx 2

2 3 5

0

14 1 = 10

3 5V x x x

3136

= units15

V

VIII. About x=h or y=k– General Case:

A.) Around the y=k:

B.) Around the x=h:

2 2( ) ( )

b

a

V k f x k g x dx

2 2( ) ( )

d

c

V h f y h g y dy