Arc Length and Surface Area of Revolution

What You’ll Learn

In this lesson you’ll learn two applications of integration: finding the length of a curve and finding the surface area when a curve is rotated around an axis.

The Concept

Arc length

The length of a curve y = f(x) from x = a to x = b is

$$L = \int_a^b \sqrt{1 + \bigl[f'(x)\bigr]^2}\,dx$$

The idea: zoom in on a tiny piece of the curve. It looks like a straight line segment with horizontal run dx and vertical rise f’(x) dx. By the Pythagorean theorem, the length of that tiny segment is sqrt(dx² + (f’(x) dx)²) = sqrt(1 + [f’(x)]²) dx. Add up all the tiny segments and you get the integral.

Surface area of revolution

When you rotate y = f(x) around the x-axis, the surface area is

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

Each tiny arc segment traces out a thin band (like a ring) when rotated. The band has circumference 2 pi f(x) and width equal to the arc length element. Multiply and integrate.

Worked Example

Example 1: Arc length of y = x^(3/2)

Find the arc length of y = x^(3/2) from x = 0 to x = 4.

First find the derivative: f’(x) = (3/2) x^(1/2).

$$L = \int_0^4 \sqrt{1 + \left(\frac{3}{2}\sqrt{x}\right)^2}\,dx = \int_0^4 \sqrt{1 + \frac{9x}{4}}\,dx$$

Let u = 1 + 9x/4, so du = 9/4 dx, giving dx = 4/9 du. When x = 0, u = 1. When x = 4, u = 10.

$$= \frac{4}{9}\int_1^{10} \sqrt{u}\,du = \frac{4}{9}\left[\frac{2}{3}u^{3/2}\right]_1^{10} = \frac{8}{27}\left(10^{3/2} - 1\right) = \frac{8}{27}\left(10\sqrt{10} - 1\right)$$

Example 2: Surface area of y = x rotated around the x-axis

Find the surface area when y = x from x = 0 to x = 1 is rotated around the x-axis. This creates a cone.

f’(x) = 1, so sqrt(1 + 1²) = sqrt(2).

$$S = 2\pi \int_0^1 x\sqrt{2}\,dx = 2\pi\sqrt{2}\left[\frac{x^2}{2}\right]_0^1 = 2\pi\sqrt{2} \cdot \frac{1}{2} = \pi\sqrt{2}$$

Drag to rotate · Scroll to zoom

The cone surface is what you get when the line y = x spins around the x-axis. The green profile curve shows the generating line.

Example 3: Surface area of y = sqrt(x) rotated around the x-axis

Find the surface area when y = sqrt(x) from x = 0 to x = 1 is rotated around the x-axis.

Drag to rotate · Scroll to zoom

This is the same trumpet shape from the volumes lesson. Here we’re calculating the surface area of the skin, not the volume inside. The green profile curve shows y = sqrt(x).

f’(x) = 1/(2 sqrt(x)), so [f’(x)]² = 1/(4x).

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

Let u = x + 1/4, du = dx. When x = 0, u = 1/4. When x = 1, u = 5/4.

$$= 2\pi\left[\frac{2}{3}u^{3/2}\right]_{1/4}^{5/4} = \frac{4\pi}{3}\left[\left(\frac{5}{4}\right)^{3/2} - \left(\frac{1}{4}\right)^{3/2}\right] = \frac{4\pi}{3}\left[\frac{5\sqrt{5}}{8} - \frac{1}{8}\right] = \frac{\pi(5\sqrt{5} - 1)}{6}$$

Real-World Application

Arc length and surface area show up whenever you need to measure curves or curved surfaces.

• In engineering, calculating the length of cable needed to follow a curved path, or the amount of sheet metal to wrap a curved surface
• In game development, arc length is used for animation timing along curved paths (so objects move at constant speed along a spline)
• In manufacturing, surface area determines how much paint, coating, or material is needed for a curved part
• In physics, surface area calculations are essential for heat transfer and fluid dynamics around curved objects

You’ve Got This

The arc length and surface area formulas both start with the same building block: sqrt(1 + [f’(x)]²). For arc length, you just integrate that. For surface area, you multiply by 2 pi times the radius (the distance from the axis). The integrals can get messy, but the setup is always the same. Many real-world arc length and surface area problems are solved numerically because the integrals don’t simplify nicely, and that’s completely fine.

Quiz

The arc length integrand contains

• A.$[f(x)]^2$
• B.$\sqrt{1 + [f'(x)]^2}$
• C.$2\pi f(x)$
• D.Just $f'(x)$

The surface area of $y = x$ from 0 to 1 rotated around the x-axis is

• A.$\pi$
• B.$2\pi$
• C.$\pi\sqrt{2}$
• D.$\pi/2$

The surface area formula for rotation around the x-axis includes the factor

• A.$2\pi x$
• B.$\pi[f(x)]^2$
• C.$2\pi f(x)$ (the circumference at each point)
• D.Just $\sqrt{1 + [f'(x)]^2}$

Arc length integrals often require

• A.Only the power rule
• B.Substitution, trig sub, or numerical methods
• C.No calculus at all
• D.Only integration by parts

In game development, arc length is used for

• A.Texture mapping only
• B.Constant-speed animation along curved paths
• C.Calculating gravity
• D.Rendering shadows