We have a parametrized curve $\gamma: \mathbb{R} \rightarrow \mathbb{R^2}$ given by $\gamma (t) = \langle e^t\cos (t), e^t\sin(t)\rangle$. I want to compute the arc-length of this curve on $[a,b]$ in general.
Is this always possible? For example, I am not sure I can do it when $a\rightarrow - \infty$. Thank you
The arc length for a $C^1$ curve on $[a,b]$ is given by $l(\gamma) = \int_a^b \|\dot{\gamma}(t)\| dt$. Since $\|\dot{\gamma}(t) \| = \sqrt{2}e^{t}$, we have $l(\gamma) = \sqrt{2}(e^{b}-e^{a})$.