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

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Answer 1 · 2017-04-20T01:32:53

#e-e^{-1}#

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.