Question: Prove that the length of a curve as computed using the arc length integral does not depend on its parametrization. More precisely, let C be

Prove that the length of a curve as computed using the arc length integral does not depend on its parametrization. More precisely, let C be the curve traced by r(t) for a ≤ t ≤ b. Let ƒ (s) be a differentiable function such that ƒ(s) > 0 and ƒ(c) = a and ƒ(d) = b. Then r₁(s) = r(ƒ(s)) parametrizes C for c ≤ s ≤ d. Verify that

S ||r' (t)|| dt = ||r(s)|| ds

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