the disk method
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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
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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
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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
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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
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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.
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Lathe
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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
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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
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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
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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
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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
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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
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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.
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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