Find arclength parametrization of a curve

Question

Given the curve :$\gamma(t)=(e^t\cdot cos(t),e^t\cdot sin(t),e^t)$, find an arclength parametrization for $\gamma$.

I calculated so far: $\vert \dot{\gamma}\vert=(e^t)^2$ and that $\gamma(t)$ is regular for $t\in[0,10\pi]$.

How do I continue?

Answer 1

$$s(t)=\int_{0}^t \|\gamma'(u)\| \ du = \int_{0}^t e^{2u} \ du = \frac{1}{2} e^{2t}$$

Now you want $g(s) = s^{-1}(t)$. We see that

$$g(s) = \log(2s)^{\frac{1}{2}}$$

is a solution. And so the arc-length parameterization is given by $\tilde{\gamma}(s)=(\gamma \circ g)(s)$.