Arc Length with Vector Functions — Question 5

Question 5

Evaluate exactly the length of $\vec r(t)=\left\langle e^t,e^{-t},\sqrt 2\,t\right\rangle$ for $0\le t\le\ln 2$. The square root in the speed appears complicated; identify its hidden perfect square.

Question 5 – Solution

Strategy Expand the squared speed and recognize $(e^t+e^{-t})^2$.

Speed $$\|\vec r'(t)\|^2=e^{2t}+e^{-2t}+2=(e^t+e^{-t})^2.$$ Both terms are positive, so $\|\vec r'(t)\|=e^t+e^{-t}$.

Length $$L=\int_0^{\ln 2}(e^t+e^{-t})\,dt =\left[e^t-e^{-t}\right]_0^{\ln 2} =\boxed{\frac 32}.$$

Insight Simplifying the norm before choosing a substitution is decisive here; treating the radical as generic would hide the elementary antiderivative.