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

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Answer 1 · 2015-12-25T19:56:18

$f(t) = (e^tln(t))/t$

$g(t) = t/(e^tln(t))$

$(df)/dt = (e^t((t-1)ln(t)+1))/t^2$

$(dg)/dt = (e^(-t)(ln(t)(1-t)-1))/ln^2(t)$

$Gamma(t) = int_2^4sqrt(((df)/dt)^2+((dg)/dt)^2)dt$

$Gamma(t) = int_2^4sqrt(((e^t((t-1)ln(t)+1))/t^2)^2 + ((e^(-t)(ln(t)(1-t)-1))/ln^2(t))^2)$

Which is $~~ 16.371300234854$

Note : Of course integral is not defined

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