# What is the arclength of (t/(5-t),8/t) on t in [1,4]?

Jul 6, 2018

$\approx 8.248414706$

#### Explanation:

we have
$x \left(t\right) = \frac{t}{5 - t}$ so $x ' \left(t\right) = \frac{5}{t - 5} ^ 2$

$y \left(t\right) = \frac{8}{t}$

$y ' \left(t\right) = - \frac{8}{t} ^ 2$

so we have to calculate

${\int}_{1}^{4} \sqrt{{\left(\frac{5}{t - 5} ^ 2\right)}^{2} + {\left(- \frac{8}{t} ^ 2\right)}^{2}} \mathrm{dt}$
this leads to an elliptic integral

with a numerical approach we get

$8.248414706$