Length of curve in Polar coordinate system

In summary, the conversation discusses calculating the length of a curve in a Polar coordinate system. The correct formula for finding the length of a curve in this system is ∫r(a)da. The conversation also explains the process of finding the length of a curve using this formula, including decomposing the curve into two parts and applying the Pythagorean theorem. The conversation concludes that for a circle, the formula simplifies to ∫rdθ.
  • #1
ltd5241
14
0
I want to caculate length of curve in Polar coordinate system like this: if r=r(a)
then length of the curve is ∫r(a)da Is this right? if not ,why ?
What's the right one ?
I konw the way in rectangular coordinate system,I just want to do it in Polar coordinate system .
 
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  • #2
You can think of it like an infinitesimal form of the Euclidean distance formula. For a function [itex]f(t)=\langle x_1(t),x_2(t),x_3(t),\ldots\rangle[/itex]

[tex]\sum_a^b \sqrt { \Delta x_1^2 + \Delta x_2^2+\Delta x_3^2+\ldots } \longrightarrow s=\int_{a}^{b} \sqrt { dx_1^2 + dx_2^2+dx_3^2+\ldots} = \int_{a}^{b} \sqrt { \left(\frac{dx_1}{dt}\right)^2 + \left(\frac{dx_2}{dt}\right)^2+\left(\frac{dx_3}{dt}\right)^2+\ldots}\text{ } dt[/tex]
 
  • #3
Is this ∫r(a)da wrong? Why?
 
  • #4
You need to know what the appropriate infinitesemal length segments are!

Now, if you do polar coordinates, you may decompose a stretch of a curve into two parts:
1. The change in the radial position from the initial point on the curve to the final point.
Infinitesemally, this has length dr.

2. Here's the tricky part: The tiny arc by which the curve segment can be approximated by a circular arc, supported by a tiny angular change between the first point and the final point on the curve.
Clearly, that circular arc lies AT a radius of the value "r", and setting the angular change as [itex]d\theta[/tex], we get the expression [itex]rd\theta[/tex] for that length segment.

3. Now, we apply the Pythogorean theorem to these two length segment to gain the proper curve segment ds:
[tex]ds=\sqrt{(dr)^{2}+(rd\theta)^{2}}[/tex]

4. Assuming that the radial position of the point of the curve is describable as a function of the angular variable, we may rewrite this as:
[tex]ds=\sqrt{(\frac{dr}{d\theta})^{2}+r^{2}}d\theta[/tex]

5. This is then the proper infinitesemal form of the lengthn segment, and the length s of the curve can then be calculated as:
[tex]s=\int_{\theta_{0}}^{\theta_{1}}\sqrt{(\frac{dr}{d\theta})^{2}+r^{2}}d\theta[/tex]

6. Note that in the case of a CIRCLE, where r is a constant function of the angle, this reduces to:
[tex]s=\int_{\theta_{0}}^{\theta_{1}}rd\theta[/tex]
as it should do.
 
  • #5
Thank you ! Especially you,Arildno.
 

What is the formula for finding the length of a curve in polar coordinates?

The formula for finding the length of a curve in polar coordinates is L = ∫√(r² + (dr/dθ)²)dθ, where r is the radius and θ is the angle.

What is the difference between finding the length of a curve in Cartesian coordinates versus polar coordinates?

In Cartesian coordinates, the length of a curve is found by using the Pythagorean theorem and integrating along the x-axis. In polar coordinates, the length is found by integrating along the angle θ and taking into account the changing radius r.

Can the length of a curve in polar coordinates ever be negative?

No, the length of a curve in polar coordinates is always a positive value. This is because the length is calculated by taking the square root of a sum of squared values, which will always result in a positive number.

How is the length of a curve affected by the shape of the curve in polar coordinates?

The length of a curve in polar coordinates is affected by the shape of the curve in terms of its radius and angle. A curve with a larger radius or a larger angle will have a longer length compared to a curve with a smaller radius or angle.

Can the length of a curve in polar coordinates be infinite?

Yes, the length of a curve in polar coordinates can be infinite if the curve has an infinite radius or angle. This can occur when the curve approaches a pole or when the angle approaches infinity.

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