Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

7.2

Disk and Washer Methods

Limerick Nuclear Generating Station, Pottstown, Pennsylvania

Find the volume of a solid of revolution using the disk method.

Find the volume of a solid of revolution using the washer method.

Find the volume of a solid with known cross sections.

Objectives

Integration as an Accumulation Process

Find the area of the region bounded by the graph of y = 4 – x2 and the x-axis. Describe the integration as an accumulation process.

Solution:

The area of the region is given by

You can think of the integration as an accumulation of the areas of the rectangles formed as the representative rectangle slides from x = –2 to x = 2, as shown in Figure 7.11.

Describing Integration as an Accumulation Process

Solution

Figure 7.11

cont’d

Volume: The Disk Method 2015

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7.2

The Disk Method

y x Suppose I start with this curve.

My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape.

So I put a piece of wood in a lathe and turn it to a shape to match the curve.

y xHow could we find the volume of the cone?

One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes.

The volume of each flat cylinder (disk) is:

2 the thicknessr

In this case:

r= the y value of the function

thickness = a small change

in x = dx

2

x dx

y xThe volume of each flat cylinder (disk) is:

2 the thicknessr

If we add the volumes, we get:

24

0x dx

4

0 x dx

42

02x

8

2

x dx

Solids of Revolution

• A solid of revolution is a solid that is generated by revolving a plane region about a line that lies in the same plane as the region; the line is called the axis of revolution. Many familiar solids are of this type.

Figure 7.15

The Disk Method

Example 1 – Using the Disk Method

Find the volume of the solid formed by revolving the region

bounded by the graph of and the x-axis

(0 ≤ x ≤ π) about the x-axis.

Solution:

From the representative

rectangle in the upper graph

in Figure 7.16, you can see that

the radius of this solid is

R(x) = f(x)

Figure 7.16

Example 1 – Solution

So, the volume of the solid of revolution is

cont’d

The region between the curve , and the

y-axis is revolved about the y-axis. Find the volume.

1x

y 1 4y

y x

1 1

2

3

4

1.707

2

1.577

3

1

2

We use a horizontal disk.

dy

The thickness is dy.

The radius is the x value of the function .1

y

24

1

1 V dy

y

volume of disk

4

1

1 dy

y

4

1ln y ln 4 ln1

02ln 2 2 ln 2

2

Find the volume of the solid formed by revolving the

region bounded by ( ) 2 and ( ) 1 about the

line 1.

f x x g x

y

1,2

1,0

b

aV A x dx

1

1A x dx

1

02 A x dx

2 22 1 1R x x x

221A x x 1 2 4

02 1 2V x x dx

1,2

1,0

1 2 4

02 1 2V x x dx

16

15

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the y-axis. 6 3 , 0, 0y x y x

8

Draw the graph. Write x in terms of y.

Volume of a cone?

The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this 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

The volume can be calculated using the disk method with a horizontal disk.

324,700,000 ft

AB/BC Homework:

Pg. 463 #1-4all, 7-10 all, 11(a and c only),12(b and d only)