Question

Find the length of the parametric curve `x = e^t + e^-t, y = 5 - 2t` from `t = 0 to t = 1`?

Answer 1

`e-e^{-1}`

Explanation:

By differentiating w.r.t. `t`,

`{dx}/{dt}=e^t-e^{-t}`

`{dy}/{dt}=-2`

Let us simplify:

`sqrt((dx/dt)^2+(dy/dt)^2)`

`=sqrt((e^t-e^(-t))^2+(-2)^2)`

`=sqrt(e^(2t)+2+e^(-2t))`

`=sqrt((e^t+e^{-t})^2)`

`=e^t+e^{-t}`

Now, we can find the arc length.

`L=\int_0^1(e^t+e^(-t))dt=[e^t-e^{-t}]_0^1=e-e^{-1}`

I hope that this was clear.