1. Chapter 5: Integration and Its Applications
5.7
Volumes of Solids of Revolution

2. Solid of Revolution Formed by revolving a plane region about a line
The line is the axis of revolution

3. The Disc Method
The volume of the solid formed by revolving the region bounded by the graph f and the x-axis (a ≤ x ≤ b) about the x-axis

4. The Disc Method

5. Example 1
Find the volume of the solid formed by revolving the region bounded by the graph of f(x) = -x2 + x and the x-axis about the x-axis.

6. The Washer Method Volume of a solid of revolution with a hole
Used with two functions
If revolved around the x-axis, we can use the Disc Method and subtract the results
f(x) is the outer radius and g(x) is the inner radius

7. The Washer Method

8. Example 2
Find the volume of the solid formed by revolving the region bounded by the graphs about the the x-axis

9. Example 3
A regulation-size football can be modeled as a solid of revolution formed by revolving the graph of
About the x-axis. Use this model to find the volume of a football.

10. Homework
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