- #1

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**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!