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Past exam of the mathematics course of the University of Oxford

2015

Prelims

Mathematics V

{c}

The <velocity vector> of the <parametric curve> is
$$
\mathbf r'(t)=(-\sin t,\cos t,\sinh t).
$$
Its <speed> is
$$
|\mathbf r'(t)|=\sqrt{1+\sinh^2t}=\cosh t,
$$
where $\cosh t>0$ fixes the sign of the square root. Hence the <arc length> is
$$
\boxed{\ell(C)=\int_{-\pi}^{\pi}\cosh t\,dt=2\sinh\pi.}
$$
The coincidence of the endpoints does not duplicate any positive <arc length>.

Solution

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

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

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

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