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A cylinder has a height of 9 feet and a volume of 706.5 cubic feet. Find the radius of the cylinder. Use 3.14 for π . AP Calculus Warm up

The disk method

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Page 1: The disk method

A cylinder has a height of 9 feet and a volume of 706.5 cubic feet.Find the radius of the cylinder. Use 3.14 for π .

AP Calculus Warm up

Page 2: The disk method

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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.

The simplest such solid is a right

circular cylinder or disk, which is

formed by revolving a rectangle

about an axis adjacent to one

side of the rectangle,

as shown in Figure 7.13.

Figure 7.13

Page 3: The disk method

3

The volume of such a disk is

Volume of disk = (area of disk)(width of disk)

= πR2w

where R is the radius of the disk and w is the width.

The Disk Method

Page 4: The disk method

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

7.3 day 2

Disk and Washer Methods

Limerick Nuclear Generating Station, Pottstown, Pennsylvania

Page 5: 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.

Page 6: The disk method

Lathe

Page 7: The disk method

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

Page 8: The disk method

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

Page 9: The disk method

This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk.

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This approximation appears to become better and better

as So, you can define the volume of the

solid as

Volume of solid =

Schematically, the disk method looks like this.

The Disk Method

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A similar formula can be derived if the axis of revolution is vertical.

Figure 7.15

The Disk Method

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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

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Example 1 – Solution

So, the volume of the solid of revolution is

cont’d

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Example 2 – Revolving about a line that is not the coordinate axis.

Find the volume of the solid formed by revolving the region bounded by: and about the line: y = 1

22)( xxf 1)( xg

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Example 1: (Use Graphing Calculator) Find the volume of the solid formed by revolving the region bounded by the graph of and the x-axis, between x = 0 and x = 3, about the x-axis.

Example 2: (No calculator) Rotate the region belowAbout the y- axis.

Example 3: (Use technology) rotate the region Bounded by the Graphs of y = 2 , and about the line y = 2

45.)( 2 xxf

44)(

2xxf

Page 16: The disk method

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

Page 17: The disk method

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

Page 18: The disk method

The region bounded by and is revolved about the y-axis.Find the volume.

2y x 2y x

The “disk” now has a hole in it, making it a “washer”.

If we use a horizontal slice:

The volume of the washer is: 2 2 thicknessR r

2 2R r dy

outerradius

innerradius

2y x

2

yx

2y x

y x

2y x

2y x

2

24

0 2

yV y dy

4 2

0

1

4V y y dy

4 2

0

1

4V y y dy

42 3

0

1 1

2 12y y

168

3

8

3

Page 19: The disk method

This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle.

The washer method formula is: 2 2 b

aV R r dx

Like the disk method, this formula will not be on the formula quizzes. I want you to understand the formula.

Page 20: The disk method

2y xIf the same region is rotated about the line x=2:

2y x

The outer radius is:

22

yR

R

The inner radius is:

2r y

r

2y x

2

yx

2y x

y x

4 2 2

0V R r dy

2

24

02 2

2

yy dy

24

04 2 4 4

4

yy y y dy

24

04 2 4 4

4

yy y y dy

14 2 2

0

13 4

4y y y dy

432 3 2

0

3 1 8

2 12 3y y y

16 64

243 3

8

3