What is the washer method?
A volume-of-revolution technique using annular cross-sections.
Washer Method Calculator
Integrate π(R(x)² − r(x)²) on [a, b] with constant or linear radius functions.
Enter radii and limits
Choose constant or linear radii, then provide coefficients and [a, b].
About this calculator
How to use Washer Method Calculator, plus the formulas and logic behind the results.
Tool description
The washer method finds volumes of solids of revolution with a hole by subtracting inner disk area from outer disk area.
How to use
- Choose constant or linear radius functions.
- Enter outer/inner radius data and interval [a, b].
- Select Calculate to view the volume.
- Ensure |R| ≥ |r| on the interval for a physical washer.
Formulas & reference
V = π ∫_a^b (R(x)² − r(x)²) dxConstant: V = π (R² − r²)(b − a)
Washer tips
- R is the outer radius from the axis
- r = 0 reduces to the disk method
- Linear radii produce a cubic integrand after squaring
- Volume is always non-negative if R² ≥ r²
Calculation logic
Integrates R² − r² exactly for constant or linear radius polynomials on [a, b], then multiplies by π.
Educational cases only—rotation about other axes may need a change of variable first.
Washer Method Calculator FAQ
Common questions about this calculator and how to use the results.
When is r = 0?
Then washers become disks: V = π∫ R² dx.
Can radii be linear?
Yes—this tool supports R = p+qx and r = u+vx.
What units?
Cube of the length unit used for radii and limits.
Does order of a and b matter?
The integral respects orientation; swap if you need positive (b−a).
Is π included?
Yes—results include the factor π numerically.