Tracing one Disk and Washer Method case
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
Working through Disk and Washer Method
Square both radii, subtract, multiply by π, and integrate across the bounds.
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.
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.
Conditions that alter Disk and Washer Method
Presenting Disk and Washer Method clearly
Washer volume adds thin cross-sectional areas π(R²−r²) along the rotation axis. Disk and Washer Method can be checked against integral core.
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.
Reviewing the Disk and Washer Method case
Before carrying Volume of revolution into another step, test Disk and Washer Method with a nearby round value for Outer radius R(x). Retaining Inner radius r(x) makes the comparison interpretable and helps separate a numerical surprise from a setup error.
Rework Disk and Washer Method without copying Volume of revolution. Begin with Outer radius R(x) and preserve Inner radius r(x). Predict the scale of the Disk and Washer Method answer, then compare it with Volume of revolution. Store a changed Lower bound as another Disk and Washer Method case.
A final Disk and Washer Method check
When Disk and Washer Method feeds later work, preserve enough digits in Volume of revolution to avoid carrying a display-rounding error forward.
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.