greg kelly, hanford high school, richland, washingtonphoto by vickie kelly, 2003 7.2 disk and washer...
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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
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
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.
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
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
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
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