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8.11 Volume with Washer Method: Revolving Around the x- or y-Axis

Part of Applications of Integration in AP® Calculus AB. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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The washer method is used when revolving a region creates circular cross sections with a hole in the middle. A washer is just a larger disc with a smaller disc removed, so its area is
π(R2−r2)\pi(R^2-r^2)
where R is the outer radius and r is the inner radius. Then the volume is found by integrating that cross-sectional area. This method is needed when the region does not touch the axis of rotation, so a hollow center appears after rotation.

The key skill is identifying the two radii correctly. Both radii must be measured from the same axis of rotation: the outer radius reaches farther from the axis, and the inner radius reaches less far. You should not just subtract the functions first and square afterward; the washer formula requires subtracting the squared radii. This lesson is very similar to the disc method, but now you account for both the outer circular area and the inner missing part.

In this unit

12 lessons

  1. 8.1Finding the Average Value of a Function on an IntervalPreview lesson
  2. 8.2Connecting Position, Velocity, and Acceleration of Functions Using IntegralsPreview lesson
  3. 8.3Using Accumulation Functions and Definite Integrals in Applied ContextsPreview lesson
  4. 8.4Finding the Area Between Curves Expressed as Functions of xPreview lesson
  5. 8.5Finding the Area Between Curves Expressed as Functions of yPreview lesson
  6. 8.6Finding the Area Between Curves That Intersect at More Than Two PointsPreview lesson
  7. 8.7Volumes with Cross Sections: Squares and RectanglesPreview lesson
  8. 8.8Volumes with Cross Sections: Triangles and SemicirclesPreview lesson
  9. 8.9Volume with Disc Method: Revolving Around the x- or y-AxisPreview lesson
  10. 8.10Volume with Disc Method: Revolving Around Other AxesPreview lesson
  11. 8.11Volume with Washer Method: Revolving Around the x- or y-AxisThis lesson
  12. 8.12Volume with Washer Method: Revolving Around Other AxesPreview lesson

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