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

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Answer 1 · 2018-07-04T14:05:33

$approx 4.003334361$

We have

$x(t)=1/(1-e^t)$
and by the power and chain rule we get

$x'(t)=-1*(1-e^(t))^(-2)(-e^t)$

$y(t)=2t$

then

$y'(t)=2$

so our integral is given by

$int_2^4sqrt((e^t/(1-e^t))^2+4)dt$
by a numerical method we get

$approx 4.003334361$

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