# What is the arclength of f(t) = (tsqrt(lnt),t^3/(4-t)) on t in [1,e]?

May 25, 2018

$\setminus \approx 16.0871$

#### Explanation:

We get
$x ' \left(t\right) = \frac{1}{2 \sqrt{\log \left(t\right)}} + \sqrt{\log \left(t\right)}$
$y ' \left(t\right) = 3 {t}^{2} / \left(4 - t\right) + {t}^{3} / {\left(4 - t\right)}^{2}$
so we have the integral
${\int}_{0}^{e} \sqrt{{\left(\frac{1}{2 \cdot \sqrt{\log \left(t\right)}} + \sqrt{\log \left(t\right)}\right)}^{2} + {\left(3 \cdot {t}^{2} / \left(4 - t\right) + {t}^{3} / {\left(4 - t\right)}^{2}\right)}^{2}} \mathrm{dt}$
Good luck!