Chapter 5: Integration and Its Applications

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Presentation transcript:

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

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

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

The Disc Method

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.

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

The Washer Method

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

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.

Homework P. 372 1-24