Arc Length

The length of a curve is the limit of the lengths of polygons that follow it. Each side of the polygon is a chord, and its length comes from the Pythagorean Theorem.

The arc length formula

Split

[a, b]

into pieces of width

\Delta x

. Over one piece the graph of

f

rises by

\Delta y

, so the chord has length

\begin{aligned} &\sqrt{(\Delta x)^2 + (\Delta y)^2} \\ &\qquad = \sqrt{1 + \left(\frac{\Delta y}{\Delta x}\right)^2}\,\Delta x. \end{aligned}

Four chords following y = 2/(1 + x²), with the Δx and Δy of one chord.

By the Mean Value Theorem, the slope of each chord equals

f'

at some point of its piece. So the polygon's length is a Riemann sum, and as

\Delta x \to 0

it approaches an integral.

A particle traveling along the graph from one end to the other covers this distance. The integrand is always at least 1, so the length is at least

b - a

Exact lengths

The square root makes most arc length integrals impossible to do by hand. One way they come out exact is when

1 + (f')^2

is a perfect square.

Example 1

Find the length of the curve

y = \frac{x^2}{8} - \ln x

from

x = 1

x = 4

The derivative is

y' = \frac{x}{4} - \frac{1}{x}

. Squaring and adding 1,

\begin{aligned} 1 + (y')^2 &= 1 + \frac{x^2}{16} - \frac{1}{2} + \frac{1}{x^2} \\[4pt] &= \frac{x^2}{16} + \frac{1}{2} + \frac{1}{x^2} \\[4pt] &= \left(\frac{x}{4} + \frac{1}{x}\right)^2. \end{aligned}

For

x > 0

the quantity in parentheses is positive, so the square root removes the square.

\begin{aligned} L &= \int_{1}^{4} \left(\frac{x}{4} + \frac{1}{x}\right)dx \\[4pt] &= \left(\frac{x^2}{8} + \ln x\right)\Big|_{1}^{4} \\[4pt] &= \left(2 + \ln 4\right) - \frac{1}{8} \\[4pt] &= \frac{15}{8} + 2\ln 2 \approx 3.261. \end{aligned}

The curve from x = 1 to x = 4, highlighted. Its length is about 3.261.

Lengths with a calculator

The perimeter of a region adds the lengths of all of its sides. Straight sides are measured directly, and each curved side is an arc length.

Length with respect to y

The formula in

x

needs

f'

to be continuous on the closed interval. If the curve has a vertical tangent at an end, write it as

x = g(y)

and integrate in

y