6'3 Volumes By Cylindrical Shells

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6.3 Volumes By Cylindrical Shells
This is an alternate method for finding the
volume of a solid of revolution that uses
cylindrical shells. The strip is parallel to the
axis of revolution. This method is sometimes
easier to use than the methods discussed in
Section 6.2.
Volume of Shell
average radius
height
thickness
2
Volume The Shell Method
h(y)
d
Typical Shell
c
r(y)
Horizontal Strip
axis of revolution
Plane Region
Volume of Typical Shell
where r(y) average radius
Solid of Revolution
h(y) height
thickness
Volume of Solid
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Examples
Use the method of cylindrical shells to find the
volume generated by rotating the region bounded
by the given curves about the specified axis.
Sketch the region and a typical shell. Example
One
about the x-axis
Example Two
about x -2
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Solutions Example One
y
y
1
r(y)
h(y)
x
x
0
1
By Shell Method
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Solutions Example Two
Points of Intersection
Vertex of parabola 1 (2, 4)
Vertex of parabola 2 (2, 8)
y
y
8
h(x)
r(x)
2
x
x
0
2
4
x -2
x -2
By Shell Method
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Example Shell Method Preferable
Find the exact volume of the solid formed by
revolving the region bounded by the curves
about the y-axis

• Methods
• Disc/Washer strip is perpendicular to axis of
  revolution
• Shell Method- strip is parallel to axis of
  revolution

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Solutions Example by Disc/Washer Method
y
(1, 2)
2
1
r
x
0
1
Note We found that the disc/washer method
requires two integrals to determine the volume of
the solid.
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Solutions Example by Shell Method
y
(1, 2)
2
1
x
0
1
Note We can see that the shell method requires
only one integral to find the volume.