SECTION 7-2Solids of Revolution

Disk Method

The Disk Method

If a region in the plane is revolved about a line, the resulting solid is a solid of revolution, and the line is called the axis of revolution.

Figure 7.13

The simplest such solid is a rightcircular cylinder or disk, which isformed by revolving a rectangleabout an axis adjacent to oneside of the rectangle

To see how to use the volume of a disk to find the volume of a general solid of revolution, consider a solid of revolution formed by revolving the plane region in Figure 7.14 about the indicated axis.

Figure 7.14

The Disk Method

How do you find the Volume of a solid generated by revolving a given area about an axis?

Slice the volume into many, many circular disks

Then add up the volume of all the disks

Solids of Revolution: Disk Method• The volume of a solid may be found by finding

the sum of the disks. • The volume of each circular disk is the area of a

circle times the width of the disk.

• Volume is found by integration. The radius of each disk is the function for each value in the interval. The width is dx

b

a

dxxfV 2))((

wxRn

1

2))((

Find Volume using Disk Method

b

a

dxxfV 2))(( d

c

dyyfV 2))((

• Revolve about a horizontal axis

• Slice perpendicular to axis – slices vertical

• Integrate in terms of x

• Revolve about a vertical axis

• Slice perpendicular to axis – slices horizontal

• Integrate in terms of y

Video clip on disk method

http://www.youtube.com/watch?v=1CbZlM09zF8 

Find Volume of a solid generated by revolving the given area about the x-axis1) Consider the function

on the interval [0,2]xxf )(

A xdx0

2

1) (Continued) Find the volume of the solid bounded by and the x-axis rotated about the x-axis on the interval [0,2]

V x 2

dx0

2

2

0

xdxV

xxf )(

2

0

2

2

xV

2)02( V

2) Find the volume of the solid generated by revolving the region bounded by y = x – x2 and y = 0 about the x - axis

1

.25

1

0

543

523

xxx

V

1

0

2dxxxV 2

1

0

432 2 dxxxxV

305

1

2

1

3

1 543

V

3) Find the volume generated by revolving the

region bounded by y = sec(x), and

y = 0 about the x - axis

2

)0tan(4tan2

V

4,

4

xx

40))(tan(2

xV

4

4

2))(sec(

dxxV

4

0

2 )(sec2

dxxV

4

4

4) Find the volume generated by revolving the

region bounded by about

the y - axis

4

81V

23yx

3

0

22

3dyyV

3 and ,0,32

yxxy

3

0

4

4

yV

Need in terms of x = ? Since revolution is about y

3

0

3dyyV

The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of the following equation revolved about the y-axis:

2.000574 .439 185x y y x

y

500 ft

500 22

0.000574 .439 185 y y dy

5) Find the volume using the disk method with a horizontal disk.

V=

24,732,567 ft 3

Assignment

Page 465 # 1-4, 7-10, 11a, 12b, 23, 25, 27, 31, and 33