Surface Area of Revolution Calculator
Compute the surface area swept out when the graph of f(x) is rotated a full turn around the x-axis over the interval [a, b].
Function f(x)
Revolving x² about the x-axis from 0 to 1
How does this calculator work?
The surface area of revolution about the x-axis is S = ∫_a^b 2π·|f(x)|·√(1 + f′(x)²) dx. This tool differentiates f numerically with a central difference and integrates with composite Simpson's rule on 1000 panels, returning the swept surface area for your chosen function and limits a and b.
Formula
How this is calculated
Pick a function f(x) and the integration limits a and b. When you revolve the curve y = f(x) about the x-axis, each point traces a circle of radius |f(x)|, and an infinitesimal arc length ds = √(1 + f′(x)²) dx sweeps a thin frustum of lateral area 2π·|f(x)|·ds. Integrating gives the total surface area S = ∫_a^b 2π·|f(x)|·√(1 + f′(x)²) dx.
The derivative f′(x) is estimated numerically with a symmetric central difference, f′(x) ≈ (f(x+h) − f(x−h)) / (2h) with h = 1e-6, which is accurate to second order. The definite integral is then evaluated with composite Simpson's rule on n = 1000 subintervals, which has error O(h⁴) and handles smooth integrands very precisely. If you enter a > b the limits are swapped so the area stays positive.
The absolute value |f(x)| ensures the radius is non-negative even where f dips below the axis. Results assume f is continuous and differentiable on [a, b]; sharp corners or vertical tangents (for example √x at x = 0) can reduce accuracy slightly. Units of the result are the square of the units used for x and f.
Frequently asked questions
If f(x) = 0 across the interval (for example sin(x) revolved over a symmetric region) the radius is zero, so no surface is generated. Choose limits where |f(x)| is positive.
This calculator revolves the curve about the x-axis. For revolution about the y-axis you would integrate 2π·|x|·√(1 + f′(x)²) dx instead.
Composite Simpson with 1000 panels plus a high-order central-difference derivative gives many correct digits for smooth functions; expect small errors only near singularities or vertical tangents.
Also known as
TG we-Calculate Editorial Team. (2026). Surface Area of Revolution Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/surface-area-of-revolution-calculator
TG we-Calculate Editorial Team. "Surface Area of Revolution Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/surface-area-of-revolution-calculator.
TG we-Calculate Editorial Team, "Surface Area of Revolution Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/surface-area-of-revolution-calculator
@misc{wecalculate_surface_area_of_revolution_calculator, title = {Surface Area of Revolution Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/surface-area-of-revolution-calculator}}, year = {2026}, note = {TG we-Calculate} }
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