(adsbygoogle = window.adsbygoogle || []).push({}); Reparametrize the curve R(t) in terms of arc length measured from the point where t = 0

R(t) is defined by x = e^{t}, y = [itex]\sqrt{2}[/itex]t, z = -e^{-t}

Arc length S = ∫ ||R'(t)||dt

||R'(t)||= sqrt{[itex]\dot{x}[/itex]^{2}+ [itex]\dot{y}[/itex]^{2}+ [itex]\dot{z}[/itex]^{2}}

The attempt at a solution

Getting R'(t) ==> x = e^{t}, y = [itex]\sqrt{2}[/itex], z = e^{-t}

Then ||R'(t)|| = sqrt{e^{2t}+ 2 + e^{-2t}} = e^{t}+ e^{-t}

S = ∫(e^{t}+ e^{-t})dt from 0 to some t

So

S = e^{t}- e^{-t}

This is the point where I get stuck. How can I transform this equation into form t = ... ?

I tried taking ln of the whole equation but it doesn't seem to work.

Help please!

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# Homework Help: Reparametrize the curve in terms of arc length

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