The washer method finds the volume of a solid formed when a two-dimensional region is revolved around an axis that the region does not touch. Because the rotation leaves a hole through the solid, each thin slice looks like a washer rather than a solid disk. This method matters because it turns a complicated three-dimensional volume into an integral using cross-sectional area.
It is especially useful for regions between two curves or regions rotated around a horizontal or vertical line.
Key Facts
- Washer volume formula: V = pi ∫[a,b] (R(x)^2 - r(x)^2) dx
- For rotation around a horizontal axis using dx, R(x) and r(x) are vertical distances from the axis of rotation to the outer and inner curves.
- Outer radius R is the larger distance from the axis of rotation to the edge of the region.
- Inner radius r is the smaller distance from the axis of rotation to the edge of the region.
- Cross-sectional washer area: A(x) = pi(R(x)^2 - r(x)^2)
- If the region is described best with y as the variable, use V = pi ∫[c,d] (R(y)^2 - r(y)^2) dy
Vocabulary
- Washer method
- A volume method that adds thin washer-shaped slices formed by rotating a region with a gap around an axis.
- Outer radius
- The distance from the axis of rotation to the farthest boundary of the rotating region.
- Inner radius
- The distance from the axis of rotation to the nearest boundary of the rotating region.
- Axis of rotation
- The line around which a plane region is revolved to create a three-dimensional solid.
- Cross section
- A flat slice of a solid used to compute volume by adding many thin areas together.
Common Mistakes to Avoid
- Using R - r instead of R^2 - r^2 is wrong because washer area depends on the area of circles, so both radii must be squared before subtracting.
- Choosing radii as curve values without measuring from the axis is wrong because the radius is always a distance from the axis of rotation, not just the height of a graph.
- Switching the outer and inner radii is wrong because it can make the integrand negative even though volume must be positive.
- Using dx when slices should be perpendicular to a vertical axis is wrong because the variable of integration must match the direction of the slice thickness.
Practice Questions
- 1 Find the volume when the region between y = 4 and y = 2 from x = 0 to x = 3 is revolved around the x-axis.
- 2 Find the volume when the region bounded by y = x + 1, y = 1, x = 0, and x = 2 is revolved around the line y = -1.
- 3 A region between two curves is revolved around a horizontal line below both curves. Explain how to decide which radius is outer and which is inner before setting up the washer integral.