Arc length of a polar curve

In summary, the arc length of a polar curve cannot be accurately modeled using the formula ds = \frac{dθ}{2π}2πr = rdθ due to the fact that in general, the radial movement is comparable to the tangential movement. The correct formula is instead \int^{β}_{α}\sqrt{r^{2}+(\frac{dr}{dθ})^{2}} \ dθ, derived from Pythagoras' theorem. This is because even in cases where the angular coordinate remains constant, there is still a non-zero arc length due to radial movement.
  • #1
Bipolarity
776
2
If we divide the polar curve into infinitely thin sectors, the arc length of a single sector can be approximated by [itex] ds = \frac{dθ}{2π}2πr = rdθ[/itex]. So why can't we model the arc length of the curve as [itex] \int^{β}_{α} rdθ[/itex]

It turns out that the correct formula is actually
[itex]\int^{β}_{α}\sqrt{r^{2}+(\frac{dr}{dθ})^{2}} \ dθ [/itex]
I know how the correct formula is derived, I just can't figure out why the reasoning for the first formula is incorrect.

BiP
 
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  • #2
By Pythagoras, ds2 = (rdθ)2+dr2. You can't go ignoring the dr term in general.
 
  • #3
haruspex said:
By Pythagoras, ds2 = (rdθ)2+dr2. You can't go ignoring the dr term in general.

My question was, why can't you use the arc length of the sector?

BiP
 
  • #4
Bipolarity said:
My question was, why can't you use the arc length of the sector?

BiP
Because in general the radial movement is of comparable magnitude to the tangential movement. In the extreme case, part of the curve might move directly away from the origin, so there dθ would be zero and you would only have dr.
 
  • #5
A straight line radiating from the origin must also have an arc length formula in polar coordinates, even though the angular coordinate remains a constant along the whole curve.
 

1. What is the formula for finding the arc length of a polar curve?

The formula for finding the arc length of a polar curve is given by L = ∫√(r^2 + (dr/dθ)^2) dθ, where r is the polar function and dr/dθ is its derivative.

2. How do you calculate the arc length of a polar curve?

To calculate the arc length of a polar curve, you first need to find the polar function and its derivative. Then, plug these values into the formula L = ∫√(r^2 + (dr/dθ)^2) dθ and integrate with respect to θ.

3. Can the arc length of a polar curve be negative?

No, the arc length of a polar curve cannot be negative. It represents the distance along the curve and is always a positive value.

4. How does the arc length of a polar curve relate to its derivative?

The arc length of a polar curve is directly related to its derivative. The formula for calculating arc length includes the square root of the polar function and its derivative, which represents the changing slope of the curve at each point.

5. Are there any special cases where the formula for arc length of a polar curve is different?

Yes, there are special cases where the formula for arc length of a polar curve may be different. For example, when the polar function is a straight line, the formula simplifies to L = r, and when the polar function is a circle, the formula becomes L = θr. These special cases are important to keep in mind when calculating arc length.

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