• Support PF! Buy your school textbooks, materials and every day products Here!

Deriving the formula for arc length of a polar function

  • Thread starter Calpalned
  • Start date
  • #1
297
6

Homework Statement


Derive ∫(dr/dθ)^2 + R^2 )^0.5 dθ

Homework Equations


x = Rcosθ
y = Rsinθ

The Attempt at a Solution


Arc length is the change in rise over run, which can be found using Pythagorean's Theorem. Rise is dy/dθ while run is dx/dθ. The arc length is [(dy/dθ)^2 + (dx/dθ)^2 ]^1/2

dx/dθ = (cosθ -Rsinθ)
dy/dθ = (sinθ + Rcosθ)
dx/dθ ^2 + dy/dθ ^2 = (cos - Rsinθ)^2 + cos^2θ - 2Rsinθcosθ + R^2sin^2θ + sin^2θ + 2Rsinθcosθ + R^2cos^θ

This simplifies to [R^2(cos^2θ + sin^2θ) + sin^2 θ+cos^2θ]^2/4
Which leaves ∫√R^2 + 1 dθ
But that is not the right formula!
 

Answers and Replies

  • #2
33,270
4,966

Homework Statement


Derive ∫(dr/dθ)^2 + R^2 )^0.5 dθ

Homework Equations


x = Rcosθ
y = Rsinθ

The Attempt at a Solution


Arc length is the change in rise over run, which can be found using Pythagorean's Theorem.
No, not at all. The arc length is the length along the curve between two points (r1, θ1) and (r2, θ2). You can approximate this length using the chord between these two points.
Calpalned said:
Rise is dy/dθ while run is dx/dθ. The arc length is [(dy/dθ)^2 + (dx/dθ)^2 ]^1/2

dx/dθ = (cosθ -Rsinθ)
dy/dθ = (sinθ + Rcosθ)
dx/dθ ^2 + dy/dθ ^2 = (cos - Rsinθ)^2 + cos^2θ - 2Rsinθcosθ + R^2sin^2θ + sin^2θ + 2Rsinθcosθ + R^2cos^θ

This simplifies to [R^2(cos^2θ + sin^2θ) + sin^2 θ+cos^2θ]^2/4
Which leaves ∫√R^2 + 1 dθ
But that is not the right formula!
 
  • #3
954
117
dx/dθ = (cosθ -Rsinθ)
dy/dθ = (sinθ + Rcosθ)
This is certainly not correct. Could you check your differentiation?
 
  • #4
297
6
x = Rcosθ
y = Rsinθ

R is a constant, so by the multiplication rule of derivatives,
dx/dθ = (1)cosθ + R(-sinθ) = cosθ - Rsinθ
dy/dθ = (1)sinθ) + R(cosθ) = sinθ + Rcosθ

I still get the same differentiation.
 
  • #5
Dick
Science Advisor
Homework Helper
26,258
618
x = Rcosθ
y = Rsinθ

R is a constant, so by the multiplication rule of derivatives,
dx/dθ = (1)cosθ + R(-sinθ) = cosθ - Rsinθ
dy/dθ = (1)sinθ) + R(cosθ) = sinθ + Rcosθ

I still get the same differentiation.
R is not a constant, it's a function of theta. And even if it were, the derivative wouldn't be 1.
 

Related Threads on Deriving the formula for arc length of a polar function

Replies
4
Views
4K
Replies
5
Views
1K
  • Last Post
Replies
2
Views
782
Replies
3
Views
2K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
1
Views
4K
  • Last Post
Replies
10
Views
6K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
7
Views
8K
Replies
6
Views
3K
Top