Question

Multivariable Calculus: Arc length parametrization of a curve; question in reply

Answer 1

Consider the curve parameterized by \[r(t) = (e^{t}, \sqrt{2}t, e^{-t}), t \ge 0\]
Find an arc length parametrization of the curve.

Answer 2

I've been able to find the arc length, \[s = e^{t} - e^{-t}\]
I hope this is right. Anyways, I believe I have to solve for t now, but I'm really stuck on this part. Any hints would be great, thanks!

Answer 3

Find R'(T)
then find |r'(t)|
arc length =\[\int\limits_{a}^{b} |(r't)| dt\]

Answer 4

use this formula when a curve is given in parametric form

Answer 5

kay, thanks I'l try that!

Answer 6

That's the wrong arc length parametrization.

Answer 7

well i've already found the arc length
or was I wrong?

Answer 8

Oh okay

Answer 9

From the given r(t) we get
\[ r'(t)=(e^t,\sqrt{2},-e^{-t}) \]
The arc length parametrization is now:
\[ s(t)=\int{||r'(t)||dt} \]

Answer 10

I'm still getting

Answer 11

\[s(t) = e^{t} - e^{-t}\]
so is that the arc length parametrization?

Answer 12

Ah, sorry, yeah, that's right, I used hyperbolic functions when I integrated so my answer looked different.

Answer 13

I got \( s(t)=2\sinh t \).

Answer 14

Which is the same

Answer 15

Okay! Thanks!
So would that be my answer? I thought that the arc length par'n was found by plugging the inverse of s(t) into r. So I'd have to solve for t, if that makes sense.

Answer 16

You're right, s(t) of course gives the arc length for a given t. And we want to put in a length and get out the corresponding point at that arc length of the curve.
The inverse is just t(s) = arcsinh(s/2), just plug that into the original parametrization.

I think I should go get some sleep, it's 4 am here. Good luck!

Answer 17

Oh wow, thanks for helping when it's so 4am where you are! Goodnight and thanks again

Answer 18

No problem, you basically solved it without any help anyways =)