Calculator Inputs
Enter cubic polynomial coefficients for the outer and inner radii. The tool evaluates the washer formula numerically and plots radius behavior across the interval.
Example Data Table
Example setup: R(x) = x + 2, r(x) = 0.5x + 0.5, interval [0, 2].
| x | Outer Radius R(x) | Inner Radius r(x) | Washer Area π(R² − r²) |
|---|---|---|---|
| 0.00 | 2.00 | 0.50 | 11.7810 |
| 0.50 | 2.50 | 0.75 | 17.8680 |
| 1.00 | 3.00 | 1.00 | 25.1327 |
| 1.50 | 3.50 | 1.25 | 33.5758 |
| 2.00 | 4.00 | 1.50 | 43.1969 |
For this example, the exact volume is 16.5π ≈ 51.8363 cubic units.
Formula Used
Washer method:
V = π ∫ab [R(t)2 − r(t)2] dt
R(t) is the outer radius and r(t) is the inner radius.
Each washer area equals π(R² − r²).
The total volume is the integral of these washer areas across the selected interval.
This calculator models:
- Outer radius as a cubic polynomial.
- Inner radius as a cubic polynomial.
- Volume using Simpson or trapezoidal numerical integration.
- Cross-sectional behavior through a plotted washer-area curve.
How to Use This Calculator
- Choose the integration variable and type an axis label for reporting.
- Enter the lower and upper bounds of the interval.
- Select the number of slices and your preferred numerical method.
- Type the outer radius coefficients for R(t).
- Type the inner radius coefficients for r(t).
- Set the linear unit label, such as cm, m, or ft.
- Press the calculate button to show the result above the form.
- Review the graph, sample rows, and export the data as CSV or PDF.
FAQs
1. What does the washer method calculate?
It finds the volume of a solid of revolution with a hollow center. Each cross-section is a washer, and the total volume is the sum of all washer areas across the interval.
2. Why are outer and inner radii both required?
A washer has thickness and a hole. The outer radius measures the full disc, while the inner radius removes the hollow part from each rotating cross-section.
3. When should the inner radius be zero?
Use zero when the solid has no hollow core at a given location. In that case, the washer becomes a full disc there.
4. Why can changing the slice count affect the answer?
This page uses numerical integration. More slices usually improve the approximation, especially when the radius functions curve sharply or change quickly over the interval.
5. Does this page find exact symbolic antiderivatives?
No. It evaluates the washer integral numerically from your coefficients and interval. The method is practical for estimation, checking, and classroom comparison.
6. Can I use it for rotation around either main axis?
Yes. The axis label is included for interpretation and reporting. Since you enter radii directly, the calculation depends on the radii and interval you supply.
7. What happens if the inner radius becomes larger?
The calculator swaps the effective radii at those points and shows a warning. That keeps each washer physically meaningful and avoids negative cross-sectional area.
8. Which units should I enter?
Use one consistent linear unit for both bounds and radii. The result will then appear in the corresponding cubic unit, such as cm³, m³, or ft³.