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
2
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
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
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
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.
Lathe
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
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.
10
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
11
A similar formula can be derived if the axis of revolution is vertical.
Figure 7.15
The Disk Method
12
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
13
Example 1 – Solution
So, the volume of the solid of revolution is
cont’d
14
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
15
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
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
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
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
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.
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
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