Finding the Arc Length Parameterization of a Vector Function

[PsychonautQQ]

Homework Statement

Find the arc length parameterization of r(t) = <(e^t)sin(t),(e^t)cos(t),10e^t>

The Attempt at a Solution

so I guess i'll start by taking the derivative of r(t)...
r'(t) = <e^t*cos(t) + e^t*sin(t), -e^t*sin(t) + e^t*cos(t), 10e^t>

ehh...
now do I do
ds = |r'(t)|dt

and integrate? what then? I don't really understand the question or what I'm trying to do really...

 

[PsychonautQQ]

okay so I found the magnitude of r'(t) and it came out to sqrt(102)*e^t .. integrate with respect to t it stays the same thing.

so S = sqrt(102)*e^t now what?

 

[Dick]

okay so I found the magnitude of r'(t) and it came out to sqrt(102)*e^t .. integrate with respect to t it stays the same thing.

so S = sqrt(102)*e^t now what?

Well, then you are done. It would sort of help if you understood the reasons for what you are doing. |dr/ds|=|dr/dt|*|dt/ds|. An arclength parametrization has |dr/ds|=1. You found |dr/dt| to be sqrt(102)e^t. So an arclength parametrization s is a solution to ds/dt=|dr/dt|=sqrt(102)e^t. You just solved that differential equation.