Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics V / 3 / a / ii

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- Solution ii

Solution

The velocity vector of the parametric curve is

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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