6'2 Volumes

1
6.2 Volumes
If a region in the plane is revolved about a
line, the resulting solid is called a solid of
revolution, the line is called the axis of
revolution.

• The Disc Method
• The Washer Method

the strip is perpendicular to the axis of
revolution
2
A. Volume The Disc Method
Vertical Strip
Typical disc
Plane Region
r
Axis of Revolution
xa
x b
Volume of Typical Disc
Solid of Revolution
where r radius
Volume of Solid
3
B. Volume The Washer Method
Typical Washer
Vertical Strip
Plane Region
Solid of Revolution
Axis of Revolution
a
b
Volume of Typical Washer
where
outer radius
inner radius
Volume of Solid
4
Examples
Find the exact volume of the solid obtained by
rotating the region bounded by the given curves
about the specified line. Sketch the region, the
solid, and a typical disc or washer.
Example One
about the y-axis
Example Two
about
5
Solutions Example One
Graph is a parabola with vertex of (1/4,1/2)
y
y
By disc method
1
(1/4,1/2)
r
x
x
0
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Solutions Example Two
y
y
x
x
1
3
By washer method