Area Between Curves and Rotation Volume
Calculate geometric area between two quadratic profiles or washer volume around the x-axis. Split crossing curves for area and validate inner and outer radii for rotation.
YOUR CALCULATION
Result
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Calculation steps
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How to calculate
For area, locate profile crossings and integrate the absolute difference on each resulting interval. For rotation, require f(x)≥g(x)≥0 across the entire interval and integrate π[f(x)²−g(x)²]. The latter subtracts the inner hole from every circular slice.
A=∫|f(x)−g(x)| dx; V=π∫[f(x)²−g(x)²] dx about the x-axis
Worked example
For f(x)=−x²+2x and g(x)=0 on [0,2], the enclosed area is 4/3 square units. Rotating the same region about the x-axis gives 16π/15≈3.351032 cubic units. For f(x)=x and g(x)=0 on [−1,1], the geometric area is 1 even though the signed integral is zero.
Assumptions and input details
Enter 1–3 coefficients in descending powers, including zero coefficients for missing powers; for example 1,0,−1 means x²−1. Coefficients accept fractions/decimals up to six places and magnitude 1e6, with reduced denominator at most 1e6. Increasing x bounds must be within ±1000. All coordinates use one chosen length unit; changing it reinterprets the numbers. Output area squares and volume cubes the conversion factor. Rotation uses only nonnegative, ordered radius profiles and the x-axis. Profile crossings and area are numerical approximations; narrow or highly curved features may be visually unresolved in the sampled chart.
References: OpenStax washer method
Unit definitions
Unit selection is independent of language. Each selector contains only units for its physical quantity. Calculations use defined base units internally.
Choose from 23 length definitions, including explicitly labelled regional measures. Area conversions square the length factor; geometric volumes cube it.
Regional measures are not interchangeable merely because their names look similar. This is a documented selection, not a claim to include every historical unit.
NIST measurement tables · Japanese traditional units · Mainland China and Hong Kong unit distinctions
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Radius-domain validation and polynomial volume coefficients use exact fractions. Final volume includes π. Crossings, geometric area, graphs and decimal displays are approximate.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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