Circumference times slant length
A tiny curve segment of length ds sweeps a narrow band whose area is approximately its circumference 2πr times ds. Integration adds those bands.
Surface Area of Revolution Calculator
Calculate the area of the surface formed when y=f(x) or a parametric curve x(t), y(t) rotates around any horizontal or vertical line. The tool differentiates the curve, builds S=2π∫ radius·ds, evaluates it numerically, and graphs the setup locally in your browser.
Enter a generating curve
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Surface area result
Calculated curved surface area
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Enter a curve, axis, and bounds to calculate the surface area.
The surface-area integral will appear here.
Calculated derivatives will appear here.
The result is a numerical approximation for a smooth generating curve that stays on one side of the axis.
Generating curve and axis
Calculate a surface area to draw the generating curve and axis of rotation.
Calculator methodology and limits
- Developed and reviewed by
- Starlight Robotics calculator team
- Last reviewed
- Method
- Parse the curve → differentiate symbolically → calculate radius and arc-length speed → scan for invalid values or axis crossings → integrate with adaptive Simpson subdivision.
- Accuracy
- The result includes the quadrature error estimate from the final subdivisions. It is an estimate, not a proof of exact rounding error.
- Input limits
- Real-valued expressions of at most 300 characters and finite values between −10¹² and 10¹². The generating curve must be smooth and remain on one side of the axis.
- Privacy
- Your equations, bounds, graph, copied summary, and downloaded samples remain on your device.
How to use the surface area of revolution calculator
- Choose Function for
y=f(x), or Parametric forx(t)andy(t). - Select a horizontal axis
y=cor vertical axisx=c, then enterc. Use zero for a coordinate axis. - Enter the curve and its finite
x- ort-interval. Keep the intended generating arc on one side of the rotation axis. - Calculate, then inspect the radius, derivative, integral, numerical error estimate, and setup graph. Copy the summary or download sampled values as CSV if useful.
What the calculation means
Shifted axes
For y=c, radius is |y−c|. For x=c, it is |x−c|. The absolute value makes distance nonnegative.
Parametric curves
Parametric mode uses ds=√(x′(t)²+y′(t)²)dt, which is useful for circles, ellipses, and curves that are awkward to express as one function.
Curved area only
The formula measures the surface swept by the curve. It does not automatically add circular end caps, bases, or any other boundary surfaces.
Worked examples
Cone from y=x on [0,1]
About y=0, radius is x and ds=√2 dx. Thus S=2π∫₀¹x√2 dx=π√2≈4.442882938. This is the lateral area only.
Unit sphere from a semicircle
Use x(t)=sin(t), y(t)=cos(t), 0≤t≤π, about x=0. Radius is sin(t) and speed is 1, so S=2π∫₀^π sin(t)dt=4π.
Cylinder from y=2 on [0,3]
About y=0, radius is 2 and ds=dx. The curved area is S=2π∫₀³2 dx=12π; the two circular ends are excluded.
Surface area of revolution formulas
| Curve and axis | Radius | Surface area |
|---|---|---|
y=f(x) about y=c | |f(x)−c| | S=2π∫ₐᵇ |f(x)−c|√(1+[f′(x)]²) dx |
y=f(x) about x=c | |x−c| | S=2π∫ₐᵇ |x−c|√(1+[f′(x)]²) dx |
x=x(t), y=y(t) about y=c | |y(t)−c| | S=2π∫ₐᵇ |y(t)−c|√(x′(t)²+y′(t)²) dt |
x=x(t), y=y(t) about x=c | |x(t)−c| | S=2π∫ₐᵇ |x(t)−c|√(x′(t)²+y′(t)²) dt |
The calculator expects finite continuous derivatives on the full entered interval and a curve that generates the intended surface once. If different parts sweep the same surface patch, the direct integral counts the patch more than once.
Reference: OpenStax Calculus Volume 2, §2.4 derives surface area from rotating a curve, and Calculus Volume 2, §7.2 gives the parametric form.
Surface area of revolution calculator FAQs
What is the general surface area of revolution formula?
The compact formula is S=2π∫r ds, where r is perpendicular distance to the rotation axis and ds is the curve’s arc-length element.
How do I rotate y=f(x) around the x-axis?
Choose a horizontal axis and enter axis value 0. The calculator uses r=|f(x)| and ds=√(1+[f′(x)]²)dx.
How do I rotate around y=2 or x=−1?
Choose horizontal for y=2 or vertical for x=−1, then enter 2 or −1 as the axis value. Radius is automatically calculated as the perpendicular distance to the shifted line.
Does the result include end caps or bases?
No. It is the area swept by the entered curve only. Add the areas of any required disks, annuli, or other end surfaces separately.
Why can’t the generating curve cross the axis?
Parts on opposite sides may rotate through the same locations and create overlapping surface patches. To avoid a misleading double count, this calculator asks you to restrict the generating curve to one side of the axis.
Why is the area unchanged when I reverse the bounds?
Surface area is geometric and nonnegative. Reversing the bounds changes only the direction in which the generating curve is traced, so the calculator orders them before integrating.
Can I use a curve with a corner or vertical tangent?
The numerical engine expects finite derivatives throughout the entered interval. Split a piecewise-smooth curve at corners or singular points. For a circle with vertical tangents, parametric mode is usually more reliable.
Does my curve data leave the browser?
No. Parsing, differentiation, integration, graphing, copying, and file preparation happen locally. A permalink contains only the inputs you explicitly choose to put in the URL.