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Arc Length Parametrization

  • #1
784
11

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...
 
Last edited:

Answers and Replies

  • #2
784
11
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?
 
  • #3
Dick
Science Advisor
Homework Helper
26,258
618
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.
 
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