Calculus I · Unit 3B · lesson

Arc Length

Concept

Learning objectives

Derive and apply the arc-length formula for graphs in x or y.

Arc Length

Explanation

Curved length is a limit of straight lengths

To measure a curve, approximate it by many short line segments. Each segment is the hypotenuse of a tiny right triangle with horizontal change $\Delta x$ and vertical change $\Delta y$ . The Pythagorean theorem gives the segment length, and the derivative replaces the ratio $\Delta y/\Delta x$ in the limiting process. This produces the arc-length factor $\sqrt{1+[f'(x)]^2}$ .

Arc-length integrals are often harder to evaluate than they are to set up. The square root may not simplify to an elementary antiderivative, so a numerical answer can be the appropriate final result. Distinguish setup from evaluation: a correct exact integral already represents the length. Check the special case of a straight line, where the formula should agree with the ordinary distance formula.

A short piece of curve behaves approximately like the hypotenuse of a tiny right triangle:

\Delta s\approx\sqrt{(\Delta x)^2+(\Delta y)^2}.

Dividing by $\Delta x$ and passing to the limit gives

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

Worked example

Length of a straight line

For $f(x)=2x+1$ on $[0,3]$ , $f'=2$ , so

L=\int_0^3\sqrt5\,dx=3\sqrt5.

This matches the distance formula between $(0,1)$ and $(3,7)$ .

Interactive checku3b-arc-01

Find the arc length of $y=2x+1$ on $[0,3]$ .

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Show hint

The derivative is constant 2.

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**Visual study**
Polygonal approximation to arc length. Show a curve approximated by many short chord segments.

After the explanation

Use the section idea

Reading lens

Distinguish geometric size from weighted amount: arc and surface formulas stretch local distance, while density assigns unequal mass to equal pieces.

Mental model

Arc length adds short chord lengths; surface area adds narrow bands; mass adds density-times-size; moments add position-weighted mass.

Decision

Identify the local size element first, then multiply by circumference, density, or position only when the modeled quantity requires it.

Common trap

Using geometric midpoint for a nonuniform object, forgetting the surface radius, or treating differential legs as independent finite lengths.

Check yourself

Can you explain why every factor belongs in the slice contribution and why the result has the intended units?

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.