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

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Answer 1 · 2018-06-10T10:35:55

$approx 10.37224180$

We have

$x(t)=e^(-2t)-t^2$
then
$x'(t)=-2^(-2t)-2t$ +

$y(t)=t-t/e^(t-1)$
then

$y'(t)=1-e^(1-t)+e^(1-t)t$
so we get the integral
$int_(-1)^1sqrt((-2e^(-2t)-2t)^2+(1-e^(1-t)+e^(1-t)t)^2)dt$
By a numerical method we get

$approx10.37224180$

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