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.
Understanding Calculus: Volume by the Washer Method
A useful way to picture a washer is to imagine a thin slice of the region before it rotates. The slice has two edges at different distances from the rotation line. When it spins, the farther edge traces a large circle and the nearer edge traces a smaller circle.
The material fills the ring between those circles. Its area comes from the area of the large circle minus the area of the small circle.
This subtraction is essential. If the inner circle is not removed, the calculation wrongly treats empty space as solid material.
Start every problem by drawing the region, the rotation line, and one representative slice. Then decide whether slices should be vertical or horizontal. A slice must be perpendicular to the rotation line to create a washer.
For a horizontal rotation line, vertical slices are often convenient, so the thickness is a tiny change in the horizontal direction. For a vertical rotation line, horizontal slices often work better. The radius is always a distance, never simply a curve equation copied into a formula.
Find each radius by measuring from the rotation line to the correct boundary. Distances must be positive, even when the rotation line lies below zero or to the left of zero.
The most common difficulty is identifying which boundary gives the outside and which gives the hole. This can change from one part of the region to another. A graph may have curves that cross, or a rotation line may lie between parts of the region.
In those cases, one integral over the full interval may not work. Split the volume into separate intervals wherever the upper boundary, lower boundary, outer edge, or inner edge changes. Test a sample slice in each interval.
Mark its two distances directly on the graph. This simple sketch prevents many errors involving reversed radii or a negative area.
Squaring the radii has an important effect. A radius twice as large produces a circle with four times the area, not twice the area. This is why a small mistake in a radius can create a large mistake in the final volume.
Keep the parentheses around each radius before squaring, especially when the rotation line is shifted away from an axis. Washer calculations appear in designs for pipes, hollow containers, drilled cylinders, rings, and parts made on a lathe.
In class, focus less on memorizing a template and more on translating the picture into distances. A correct graph, slice direction, and pair of radii usually make the integral straightforward.
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.