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AP Calculus AB: 10.6.4 The Washer Method across the x-Axis

Mathematics8 CardsCreated 3 months ago

This set explains how to find the volume of solids of revolution with holes using the washer method. It highlights the importance of subtracting the inner radius area from the outer radius area for each cross-sectional slice, setting up integrals with proper radii, and solving volume problems involving regions not flush with the axis of rotation.

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The Washer Method across the x-Axis

•Using the disk method, the volume V of a solid of revolution is given by , where R(x) is the radius of
the solid of revolution with respect to x.
• Washers are disks with smaller disks removed from the center.

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Key Terms

Term
Definition

The Washer Method across the x-Axis

•Using the disk method, the volume V of a solid of revolution is given by , where R(x) is the radius of
the solid of revolution with respect to ...

note

  • Suppose you are asked to find the volume of the
    solid of revolution described on the left.

  • The first step would be to gra...

What is the volume of the solid of revolution generated by revolving the area bounded by y = 4, y = 2x, and x = 1 around the x‑axis?

20π/3 units^3

What is the volume of the solid of revolution generated by revolving the area bounded by y = 2, y = x + 3, and x = 4 around the x‑axis?

275π/3 units^3

What is the volume of the solid of revolution generated by revolving the area bounded by y = 4 and y = x^ 2 around the x‑axis?

256π/5 units^3

What is the volume of the solid of revolution generated by rotating the area bounded by y = 10, y = 8, x = −1, and x = 3 around the x‑axis?


144π units^3

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AP Calculus AB: 10.6.4 The Washer Method across the x-Axis
TermDefinition

The Washer Method across the x-Axis

•Using the disk method, the volume V of a solid of revolution is given by , where R(x) is the radius of
the solid of revolution with respect to x.
• Washers are disks with smaller disks removed from the center.

note

  • Suppose you are asked to find the volume of the
    solid of revolution described on the left.

  • The first step would be to graph the region in question. The region is bound by a parabola and three lines.

  • Notice that the region is not flush with the axis of revolution. This means that there will be a hole in the solid of revolution.

  • How can you calculate the volume of a solid of revolution with a hole in it?

  • Whenever you encounter a difficult problem, it is a good idea to consider a simpler problem first.

  • Consider an apple that has had its center cored out. What shape does the cross-section take when you slice the solid?

  • Each slice looks like a regular apple slice, but with a hole in the center. These slices are called washers. The area of each piece is easy to calculate, since all you have to do is subtract the area of the inner circle from the area of the outer circle.

  • Finding volumes using washers is like finding volumes using the disk method. But instead of integrating the whole area of the disk, you only integrate the area of the washer. Remember, the area of the washer equals the area given by the outer radius minus the area given by the inner radius.

  • Now you can find the volume of solids of revolution with holes in them.

What is the volume of the solid of revolution generated by revolving the area bounded by y = 4, y = 2x, and x = 1 around the x‑axis?

20π/3 units^3

What is the volume of the solid of revolution generated by revolving the area bounded by y = 2, y = x + 3, and x = 4 around the x‑axis?

275π/3 units^3

What is the volume of the solid of revolution generated by revolving the area bounded by y = 4 and y = x^ 2 around the x‑axis?

256π/5 units^3

What is the volume of the solid of revolution generated by rotating the area bounded by y = 10, y = 8, x = −1, and x = 3 around the x‑axis?


144π units^3

What is the volume of the solid of revolution generated by revolving the area bounded by y = 2, y = x, and x = 0 around the x‑axis?

16π/3 units^3

What is the volume of the solid of revolution generated by rotating the area bounded by y = 2, y = 3, x = 0, and x = 4 around the x‑axis?

20π units^3