 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-v/3/a/ii/solution

r^{'} (t) = (- sin t, cos t, sinh t) .

Its speed is

∣ r^{'} (t) ∣ = 1 + sinh^{2} t = cosh t,

where

cosh t > 0

fixes the sign of the square root. Hence the arc length is

ℓ (C) = \int_{- π}^{π} cosh t d t = 2 sinh π .

The coincidence of the endpoints does not duplicate any positive arc length.

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