What is the shell method?
A volume technique integrating cylindrical shells around an axis.
Shell Method Calculator
Use cylindrical shells with radius x and height f(x) (constant or linear) on [a, b].
Enter height and limits
Choose constant or linear height, then enter coefficients and [a, b] (assume a ≥ 0).
About this calculator
How to use Shell Method Calculator, plus the formulas and logic behind the results.
Tool description
Cylindrical shells integrate circumference × height × thickness: 2π x f(x) dx when revolving about the y-axis.
How to use
- Select constant or linear height.
- Enter height parameters and the interval [a, b].
- Select Calculate to view the shell-method volume.
- Prefer a ≥ 0 so shell radius x stays non-negative.
Formulas & reference
V = 2π ∫_a^b x f(x) dxConstant height: V = 2π h (b² − a²)/2
Shell tips
- Shell circumference is 2πx
- Useful when washers need awkward horizontal slices
- Linear height gives ∫ (mx² + kx) dx
- Check that f(x) ≥ 0 for geometric volume
Calculation logic
Integrates x·f(x) exactly for constant or linear f, then multiplies by 2π.
Assumes revolution about the y-axis with vertical shells of radius x.
Shell Method Calculator FAQ
Common questions about this calculator and how to use the results.
Why multiply by x?
x is the shell radius from the y-axis.
Constant vs linear height?
Constant is a horizontal top; linear allows a slanted height function.
Can a be negative?
Radius |x| would be needed; this educational tool assumes a ≥ 0.
How does this compare to washers?
Same volume, different slicing direction—choose based on the functions.
Is 2π included?
Yes.