# What is the arclength of f(t) = (e^(t)-e^(t^2)/t,t^2-1) on t in [1,3]?

Jun 23, 2018

$\approx 2681.58$

#### Explanation:

With
$x \left(t\right) = {e}^{t} - {e}^{{t}^{2}} / t$ we get

$x ' \left(t\right) = {e}^{t} - 2 {e}^{{t}^{2}} + {e}^{{t}^{2}} / {t}^{2}$

$y \left(t\right) = {t}^{2} - 1$
so

$y ' \left(t\right) = 2 t$
and we have to solve

${\int}_{1}^{3} \sqrt{{\left({e}^{t} - 2 {e}^{{t}^{2}} + {e}^{{t}^{2}} / t\right)}^{2} + {\left(2 t\right)}^{2}} \mathrm{dt} \approx 2681.58$