Question

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

Answer 1

`approx 10.37224180`

Explanation:

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`