The Surface Area Calculus Calculator calculates the area of the surface formed when a curve y = f(x) is rotated around the x-axis or the y-axis between x = a and x = b.
Surface Area Calculus Calculator (Surface of Revolution)
Surface area of a solid of revolution: rotate y = f(x) on [a, b] about the x-axis or y-axis and integrate numerically.
It evaluates the integral S = 2π ∫ f(x)√(1 + f′(x)²) dx (x-axis) or S = 2π ∫ x√(1 + f′(x)²) dx (y-axis) numerically, so it works for functions that are hard to integrate by hand. Type your function and limits below.
Formula
Rotation about the x-axis:
S = 2π ∫ₐᵇ |f(x)| √(1 + f′(x)²) dx
Rotation about the y-axis:
S = 2π ∫ₐᵇ |x| √(1 + f′(x)²) dx
Where:
- S = surface area of the solid of revolution
- f(x) = the curve being rotated; f′(x) = its derivative (slope)
- a, b = the lower and upper limits of x
- √(1 + f′(x)²) dx = a small piece of arc length along the curve
How the Calculation Works
Step 1: The curve is split into tiny pieces. Each piece, when rotated, sweeps out a thin band like a slice of a cone.
Step 2: Each band’s area is its circumference (2π × distance from the axis) times its slant length (the arc length √(1 + f′²) dx).
Step 3: All the bands are added up. The calculator does this with Gauss–Legendre quadrature and finds f′(x) numerically, and it warns you if the integral does not converge.
Variables Explained
| Input | Meaning | Example |
|---|---|---|
| f(x) | Function of x | sqrt(x), x^2, 2*x, sin(x) |
| a | Lower limit | 0 |
| b | Upper limit (must be greater than a) | 1 |
| Axis | x-axis or y-axis | x-axis |
| S | Surface area | square units |
Supported syntax: + − * / ^, sqrt, sin, cos, tan, ln, exp, abs and pi. Implicit multiplication such as 2x also works.
Example Calculation
y = √x about the x-axis, 0 ≤ x ≤ 1
f′(x) = 1 ÷ (2√x), so f(x)√(1 + f′²) = √(x + ¼)
S = 2π ∫₀¹ √(x + ¼) dx = (π ÷ 6)(5√5 − 1)
Result: S ≈ 5.3304 square units
y = x² about the x-axis, 0 ≤ x ≤ 1
S = 2π ∫₀¹ x²√(1 + 4x²) dx = (π ÷ 32)(18√5 − ln(2 + √5))
Result: S ≈ 3.8097 square units
Check with a cone
Rotating y = 2x from 0 to 3 about the x-axis makes a cone with radius 6 and slant height √45. The calculator returns π × 6 × √45 ≈ 126.45, matching the cone formula πrs.
Formula Variations
Functions of y
If your curve is written as x = g(y), rotating about the y-axis uses S = 2π ∫ g(y)√(1 + g′(y)²) dy. Rewrite it as y = f(x) to use this calculator, or swap the variable names.
Parametric curves
For x(t), y(t) about the x-axis: S = 2π ∫ y(t)√(x′(t)² + y′(t)²) dt. This calculator handles y = f(x) only.
Engineering Assumptions
The integral is evaluated numerically, not symbolically, so results are decimal approximations. If f(x) or its slope becomes infinite inside the interval (for example ln(x) near 0), the calculator reports that the integral does not converge.
Units and Dimensions
The answer is in square units of whatever length unit x and f(x) use. If x is in centimeters, the surface area is in cm².
Frequently Asked Questions
What is the formula for surface area of revolution?
About the x-axis, S = 2π ∫ f(x)√(1 + f′(x)²) dx from a to b.
How do you find surface area when rotating about the y-axis?
Replace the radius f(x) with x: S = 2π ∫ x√(1 + f′(x)²) dx.
Why is there a square root in the surface area formula?
√(1 + f′(x)²) dx is the arc length of a small piece of the curve. The surface is measured along the curve, not along the x-axis.
What is the difference between volume and surface area of revolution?
Volume uses π ∫ f(x)² dx (disks). Surface area uses 2π ∫ f(x)√(1 + f′²) dx (bands).
Why does my integral not converge?
The function or its slope becomes infinite within your limits. Choose limits that avoid that point.