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

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Answer 1 · 2018-07-06T08:56:11

$approx 8.248414706$

we have
$x(t)=t/(5-t)$ so $x'(t)=5/(t-5)^2$

$y(t)=8/t$

$y'(t)=-8/t^2$

so we have to calculate

$int_1^4 sqrt((5/(t-5)^2)^2+(-8/t^2)^2)dt$
this leads to an elliptic integral

with a numerical approach we get

$8.248414706$

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