Arc Length

Level 1 - Calculus topic page in Applications Of Integrals.

Principle

Arc length measures the length of a curve. Calculus obtains it by adding tiny straight-line distances along the graph and passing to a limit.

\(s\)

arc length

\(f'(x)\)

slope of the graph y=f(x)

\(a,b\)

endpoints of the x-interval

For a differentiable graph \(y=f(x)\), use

Arc length

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

The square root comes from Pythagoras applied to a small horizontal change and the corresponding vertical change.

Question

Find the arc length of \(y=3x+1\) from \(x=0\) to \(x=2\).

Answer

Here \(f'(x)=3\). The arc length is \(\int_0^2\sqrt{1+3^2}\,dx=\int_0^2\sqrt{10}\,dx=2\sqrt{10}\).

Question

Set up the arc length of \(y=x^2\) from \(x=0\) to \(x=1\).

Answer

The derivative is \(f'(x)=2x\). Therefore the arc length is \(\int_0^1\sqrt{1+(2x)^2}\,dx=\int_0^1\sqrt{1+4x^2}\,dx\).