# What is the arclength of (1/(1-e^t),2t) on t in [2,4]?

Jul 4, 2018

$\approx 4.003334361$

#### Explanation:

We have

$x \left(t\right) = \frac{1}{1 - {e}^{t}}$
and by the power and chain rule we get

$x ' \left(t\right) = - 1 \cdot {\left(1 - {e}^{t}\right)}^{- 2} \left(- {e}^{t}\right)$

$y \left(t\right) = 2 t$

then

$y ' \left(t\right) = 2$

so our integral is given by

${\int}_{2}^{4} \sqrt{{\left({e}^{t} / \left(1 - {e}^{t}\right)\right)}^{2} + 4} \mathrm{dt}$
by a numerical method we get

$\approx 4.003334361$