Finding the Arc Length Parameterization of a Vector Function

In summary, the arc length parameterization of r(t) = <(e^t)sin(t),(e^t)cos(t),10e^t> is s = sqrt(102)*e^t. This is found by taking the derivative of r(t), finding the magnitude of r'(t), and solving the differential equation ds/dt=|dr/dt|=sqrt(102)e^t.
  • #1
PsychonautQQ
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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...
 
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  • #2
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
PsychonautQQ said:
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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1. What is Arc Length Parametrization?

Arc Length Parametrization is a mathematical concept used to describe a curve or path in a way that allows for easy calculation of the length of the curve. It involves using a parameter called arc length, which represents the distance traveled along the curve from a starting point.

2. Why is Arc Length Parametrization important?

Arc Length Parametrization is important because it allows for accurate and efficient calculation of the length of a curve, which is useful in various fields such as physics, engineering, and computer graphics. It also allows for the simplification of complex equations and makes it easier to analyze and manipulate curves.

3. How is Arc Length Parametrization calculated?

To calculate Arc Length Parametrization, the curve is first divided into small segments, and the length of each segment is approximated using a formula. The lengths of all these segments are then added together to get an approximation of the total length of the curve. As the number of segments increases, the approximation becomes more accurate.

4. What are some applications of Arc Length Parametrization?

Arc Length Parametrization has many applications, including finding the distance traveled by a moving object along a curved path, calculating the work done by a force on a curved surface, and determining the curvature of a curve. It is also used in computer graphics to create smooth and realistic curves.

5. Are there any limitations to Arc Length Parametrization?

One limitation of Arc Length Parametrization is that it can only be used for curves that can be represented by a single parameter. Also, the calculation of Arc Length Parametrization can become extremely complex for highly irregular or fractal curves. In such cases, other methods may be more suitable for determining the length of the curve.

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