Mathwords: Surface of Revolution

Key Formula

S = 2\pi \int_a^b r(x)\,\sqrt{1 + \left[f'(x)\right]^2}\;dx

Where:

$S$ = Surface area of the surface of revolution
$f(x)$ = The function whose graph is being rotated
$f'(x)$ = The derivative of f(x)
$r(x)$ = The distance from the curve to the axis of rotation (equals f(x) when rotating about the x-axis)
$a, b$ = The interval endpoints over which the curve is rotated

Worked Example

Problem: Find the surface area of the surface formed by rotating f(x) = 2x on the interval [0, 3] about the x-axis.

Step 1: Identify the function, its derivative, and the radius of rotation. Since we rotate about the x-axis, the radius r(x) = f(x) = 2x, and f'(x) = 2.

r(x) = 2x, \quad f'(x) = 2

Step 2: Set up the surface area integral using the formula.

S = 2\pi \int_0^3 2x\,\sqrt{1 + (2)^2}\;dx = 2\pi \int_0^3 2x\,\sqrt{5}\;dx

Step 3: Factor constants out of the integral and integrate.

S = 4\pi\sqrt{5}\int_0^3 x\;dx = 4\pi\sqrt{5}\left[\frac{x^2}{2}\right]_0^3 = 4\pi\sqrt{5}\cdot\frac{9}{2}

Step 4: Simplify to get the final answer.

S = 18\pi\sqrt{5} \approx 126.33

Answer: The surface area is

18\pi\sqrt{5}

square units, approximately 126.33 square units.

Another Example

This example involves a non-linear function (a square root) that requires simplification and u-substitution, unlike the first example where the integrand stayed elementary.

Problem: Find the surface area of the surface formed by rotating f(x) = \sqrt{x} on the interval [0, 4] about the x-axis.

Step 1: Identify the function, derivative, and radius. Here r(x) = √x and we need f'(x).

f(x) = \sqrt{x}, \quad f'(x) = \frac{1}{2\sqrt{x}}

Step 2: Set up the surface area integral.

S = 2\pi\int_0^4 \sqrt{x}\,\sqrt{1 + \frac{1}{4x}}\;dx = 2\pi\int_0^4 \sqrt{x}\,\sqrt{\frac{4x+1}{4x}}\;dx

Step 3: Simplify the integrand. The √x in the numerator cancels with the √x in the denominator of the radical.

S = 2\pi\int_0^4 \sqrt{x}\cdot\frac{\sqrt{4x+1}}{2\sqrt{x}}\;dx = \pi\int_0^4 \sqrt{4x+1}\;dx

Step 4: Evaluate using the substitution u = 4x + 1, so du = 4 dx. When x = 0, u = 1; when x = 4, u = 17.

S = \pi\cdot\frac{1}{4}\int_1^{17} u^{1/2}\;du = \frac{\pi}{4}\left[\frac{2u^{3/2}}{3}\right]_1^{17} = \frac{\pi}{6}\left(17^{3/2} - 1\right)

Step 5: Compute the numerical value.

S = \frac{\pi}{6}(17\sqrt{17} - 1) \approx \frac{\pi}{6}(70.09 - 1) \approx 36.18

Answer: The surface area is

\dfrac{\pi}{6}(17\sqrt{17} - 1)

square units, approximately 36.18 square units.

Frequently Asked Questions

What is the difference between a surface of revolution and a solid of revolution?

A surface of revolution is only the outer shell — the two-dimensional boundary created by spinning a curve. A solid of revolution is the entire three-dimensional region enclosed by that surface. You compute surface area for a surface of revolution and volume for a solid of revolution, using different integral formulas.

How do you find the surface area when the curve is rotated about the y-axis instead of the x-axis?

When rotating about the y-axis, the radius of rotation becomes the horizontal distance from the curve to the y-axis, which is x. The formula becomes

S = 2\pi\int_a^b x\,\sqrt{1+[f'(x)]^2}\;dx

. Alternatively, you can express x as a function of y and integrate with respect to y, using

S = 2\pi\int_c^d g(y)\,\sqrt{1+[g'(y)]^2}\;dy

.

What are common examples of surfaces of revolution?

A sphere is formed by rotating a semicircle about its diameter. A cylinder is formed by rotating a horizontal line segment about a parallel axis. A cone results from rotating a slanted line through the axis. A torus (doughnut shape) is created by rotating a circle about an axis that does not intersect it.

Surface of Revolution vs. Solid of Revolution

	Surface of Revolution	Solid of Revolution
What it is	The 2D boundary (shell) formed by rotating a curve	The 3D region (filled volume) enclosed by the rotated curve
What you calculate	Surface area	Volume
Key formula (rotation about x-axis)	$S = 2\pi\int_a^b f(x)\sqrt{1+[f'(x)]^2}\,dx$	$V = \pi\int_a^b [f(x)]^2\,dx$ (disk method)
Dimensionality of result	Square units (area)	Cubic units (volume)
Analogy	Like the skin of an orange	Like the entire orange including the inside

Why It Matters

Surfaces of revolution appear throughout calculus courses when you study applications of integration, typically right after learning about solids of revolution. Engineers and physicists use these surfaces to model real objects with rotational symmetry — satellite dishes (paraboloids), pipes (cylinders), funnels (cones), and pressure vessels (ellipsoids). Mastering this concept also builds your skill with arc-length integrals, since the surface area formula is a direct extension of the arc-length formula.

Common Mistakes

Mistake: Confusing the surface area formula with the volume (disk/washer) formula and forgetting the square root term √(1 + [f'(x)]²).

Correction: The surface area formula includes the arc-length factor √(1 + [f'(x)]²) because you are measuring the actual length of the curve as it sweeps around the axis, not just stacking flat disks. Always include this factor when computing surface area.

Mistake: Using f(x) as the radius of rotation regardless of which axis the curve is rotated about.

Correction: The radius r(x) is the perpendicular distance from the curve to the axis of rotation. For rotation about the x-axis, r(x) = f(x). For rotation about the y-axis, r(x) = x. Always identify the correct radius before setting up the integral.

Related Terms

Solid of Revolution — The filled 3D region inside a surface of revolution
Surface Area of a Surface of Revolution — The integral formula for computing its area
Surface — General term for a 2D shape in 3D space
Axis of Rotation — The line about which the curve is rotated
Coplanar — Curve and axis must lie in the same plane
Curve — The generating curve that is rotated
Plane — The flat surface containing the curve and axis
Three Dimensions — The space in which the surface exists