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
x=a
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

6. Solutions: Example Twoy y x x 1 3
By washer method