Arc Length with Vector Functions — Question 9

Question 9 – Solution

Strategy Use the identity $1+\sinh^2t=\cosh^2t$ and the positivity of $\cosh t$.

Length $$\|\vec r'(t)\|=\sqrt{1+\sinh^2t}=\cosh t, \qquad L=\int_{-a}^{a}\cosh t\,dt=2\sinh a.$$ Therefore $\boxed{L=2\sinh a}$.

Prescribed length Setting $2\sinh a=6$ gives $\sinh a=3$, hence $$\boxed{a=\operatorname{arsinh}(3)=\ln(3+\sqrt{10})}.$$

Symmetry The even speed makes the two halves have equal length; equivalently, $L=2\int_0^a\cosh t\,dt$.