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

The unit circle with the point (−1, 0) removed has parametrization (see Exercise 73 in Section 11.1)

Use this parametrization to compute the length of the unit circle as an improper integral. Hint: The expression for ∥r′(t)∥ simplifies.

The involute of a circle, traced by a point at the end of a thread unwinding from a circular spool of radius R, has parametrization (see Exercise 26 in Section 12.2)

r(θ) = 〈R(cos θ + θ sin θ), R(sin θ − θ cos θ)〉

Find an arc length parametrization of the involute.

The curve r(t) = 〈t − tanh t, sech t〉 is called a tractrix (see Exercise 92 in Section 11.1).