Surface Area Of Revolution Calculator
Work out surface area of revolution instantly with clear inputs, formula shown and shareable results.
About Surface Area Of Revolution Calculator
Rotating a curve about the x-axis sweeps a surface of area 2π∫|f(x)|·√(1 + (f')²) dx: each element is a frustum band of radius f(x) and slant width ds. Pappus's theorem recovers the mean radius as area divided by 2π times arc length.
Formula
Surface of revolution
S = 2π ∫ₐᵇ |f(x)| √(1 + (f'(x))²) dx
Worked example
With the default values below, Surface Area Of Revolution Calculator returns Surface area of revolution of 35.54306351. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Curve coefficients, highest power first
- 1, 0
- From x =
- 1
- To x =
- 3
Results
- Surface area of revolution
- 35.54306351
- Arc length of the generating curve
- 2.82842712
- Volume enclosed
- 27.22713633
- Mean radius (Pappus)
- 2.00000000
Step by step
- 1Values used
- 2Surface of revolution
- 3Surface area of revolution= 35.54306351
- 4Arc length of the generating curve= 2.82842712
- 5Volume enclosed= 27.22713633
- 6Mean radius (Pappus)= 2.00000000
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
Revolve y = x from 0 to 3 about the x-axis.
S = 2π∫x√2 dx = 9π√2 ≈ 39.9822, the lateral area of a cone with radius 3 and slant height 3√2.
Why include the √(1 + (f')²) factor?
Because the band's width follows the slanted curve, not the horizontal projection.