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8.12 Volume with Washer Method: Revolving Around Other Axes

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The washer method also extends to axes other than the x-axis or y-axis. The structure stays the same:
V=∫π(R2−r2)V=\int \pi(R^2-r^2)
but now R and r are distances from the region’s boundaries to the given axis of rotation. If the axis is y=k, then the radii are vertical distances. If the axis is x=h, then the radii are horizontal distances. This is the same geometric idea as before, but with shifted axes.

The main challenge is careful setup. You should always draw or imagine the region and identify which curve is farther from the axis and which is closer. Once that is clear, the rest is just the washer formula. Because the axis is shifted, signs and subtraction matter even more than before, but the underlying idea is still just “outer circle minus inner circle” accumulated across the interval.

In this unit

13 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-AxisPreview lesson
  12. 8.12Volume with Washer Method: Revolving Around Other AxesThis lesson
  13. 8.13The Arc Length of a Smooth, Planar Curve and Distance TraveledPreview lesson

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