Solution
= Solution
{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>.