We want to reparametrize the curve
$\displaystyle \vec{r}(t)=
Here is what I tried, but I've hit a snag:
$\displaystyle \vec{v}(t)=<3t^2, 2t, \sqrt{5}t>$ $\displaystyle |\vec{v}(t)|=3t\sqrt{t^2+1}$
$\displaystyle s=\int_{0}^{t}3t\sqrt{t^2+1}d\tau=3t^2\sqrt{t^2+1}$
I think I'm missing something here. but assuming everything is correct, we need to solve:
$\frac{s^2}{9}=t^4(t^2+1)$ for $t$, and then we are nearly done. I can't seem to solve for $t$ however, brain fart? Assuming we did, we just plug $t$ in for the expression in terms of $s$ in the original equation and we are done? P.S. this is exam review, not homework!
Thanks for reading!