CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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The scalar line integral of one is the arc length

Statement

For every piecewise- $C^{1}$ path $γ$ ,

$\int_{γ} 1 d s = L (γ) .$

Facts & Assumptions

Given: A piecewise- $C^{1}$ path $γ$ , with an admissible partition when its parameter interval is nondegenerate.

[L1]

On a nondegenerate interval, substituting $f = 1$ in the scalar line-integral definition gives the sum of the speed integrals over the smooth pieces; on a singleton interval the scalar line integral is zero (Scalar line integrals with respect to arc length and vector-field line integrals).

[L2]

That sum of speed integrals equals the path length; on a singleton interval the empty sum and the length are both zero (A continuous piecewise- $C^{1}$ path is rectifiable and its length is the sum of the speed integrals over its pieces).

Proof

technique · direct

1.1

On a nondegenerate interval, [L1] gives $\int_{γ} 1 d s = \sum_{i < m} \int_{t_{i}}^{t_{i + 1}} ∥ v_{i} (t) ∥_{2} d t .$

givenL1

2.1

By [L2], the right-hand side of step 1.1 is $L (γ)$ .

step 1.1L2

3.1

On a singleton interval both sides are zero by [L1] and [L2]. A constant path on a nondegenerate interval has zero speed, so step 2.1 gives zero on both sides there as well.

L1L2step 2.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 49 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis II, section 9.2 (standard reference, not scraped)