A worked example
Rotating y=√x from 0 to 4 about the x-axis with inner radius zero gives volume 8π.
Math calculator
Disk and Washer Method Calculator
Integrate π(R²−r²) around the x-axis. Input changes update both volume of revolution and the supporting steps.
Disk and Washer Method inputs
Complete the fields
Where Disk and Washer Method applies
It computes solids of revolution when outer and inner radii are functions of x. A related application of Disk and Washer Method is planar separation.
Reading the calculation
Washer volume adds thin cross-sectional areas π(R²−r²) along the rotation axis. Disk and Washer Method can be checked against integral core.
A correct volume of revolution therefore depends on choosing the model before entering the numbers.
Working through Disk and Washer Method
Square both radii, subtract, multiply by π, and integrate across the bounds.
Numerical limits of Disk and Washer Method
Radii must be nonnegative and outer radius cannot fall below inner radius. The expressions must measure distance from the axis.
When disk and washer method looks surprising, restore the sample and vary lower bound by itself.
Checking the model and magnitude
For this disk and washer method calculation, the labels Outer radius R(x), Inner radius r(x), Lower bound and Upper bound carry mathematical meaning. A transposed entry can remain numerically valid while describing an entirely different setup.
How the result responds
Scaling both radii by c scales volume by c² when bounds remain fixed. A small controlled input change is enough to test the expected direction.
Area-between-curves measures a planar gap; washers rotate that gap into cross-sections. Checking the requested noun is often enough to select the right model.
Recording the answer
An exact expression can preserve factors, radicals, or π that a decimal hides. Use the representation suited to the next task and label it clearly as volume of revolution.
Reproducibility here depends on the inputs more than the interface. Preserve Outer radius R(x), Inner radius r(x), Lower bound and Upper bound, the operation shown, and enough unrounded digits for the next calculation.
Reproducing this result later
Keep Outer radius R(x), Inner radius r(x), Lower bound, and Upper bound with the Disk and Washer Method output. Preserve more digits internally than the final report needs. Subtraction in finite differences and cumulative integration can lose significant digits, so rounding intermediate samples can damage the final estimate disproportionately.
For this disk and washer method result, treat the shown digits as conditional on the entered domain and sampling choices. Keeping those choices with the value helps distinguish a stable computation from a coincidental rounded match.
Questions about Disk and Washer Method
What is a disk?
A washer with inner radius zero.
Why square radii?
Cross-sectional circle area uses radius squared.
What if the axis differs?
Radii must be redefined as distances to it.