Concept preview

What you will be able to do

1. 01Derive and state the arc-length formula $L=\displaystyle\int_a^b \sqrt{1+[f'(x)]^2}\,dx$ from the Pythagorean length of an infinitesimal piece of the curve $\sqrt{dx^2+dy^2}$.
2. 02Set up the arc-length integral for a given smooth function $f$ on $[a,\,b]$, deciding whether it evaluates to a clean closed form or should be left as a set-up or numerical estimate.
3. 03Interpret arc length as the distance traveled along a curve, and connect the formula to the parametric arc-length integral introduced in Unit 9.

Take a smooth curve $y=f(x)$ running from $x=a$ to $x=b$. "Smooth" here means $f$ has a continuous derivative on the interval, so the curve has no corners or breaks — at every zoom level a small enough piece looks like a straight segment. That is exactly the property we will exploit. We cannot measure the bending curve directly, but we can measure a straight segment, and a tiny piece of a smooth curve is essentially straight.