What is Arc length: Definition and 286 Discussions

Arc length is the distance between two points along a section of a curve.
Determining the length of an irregular arc segment is also called rectification of a curve. The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases.

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  1. S

    Question about arc length and the condition dx/dt > 0

    This is not homework That passage is from James Stewart (Multivariable Calculus). I want to ask about the condition dx/dt > 0. If dx / dt < 0, the formula can't be used? Thanks
  2. Alex Schaller

    I Electric arc length in relation to voltage

    Provided the length of the arc of an electrostatic generator is 7 cm, can we state that its voltage is around 210,000 V? (details as per link below) https://photos.app.goo.gl/MKSpviQwPh9eS5jZ7
  3. Ennio

    A Calculate the exact or approximate length of one period of the wave function

    The known expression of the wave function is where A is the amplitude, k the wave number and ω the angular velocity. The mathematical definition of arc length for a generical function in an interval [a,b] is where, in our sinusoidal case: For our purpose (calculation of the length in one...
  4. A

    MHB Formula to find arc radius using arc length, chord length, and/or segment angle

    I'm attempting to write some code using the Ruby programming language that will give me the radius of an arc but the only pieces of information I have to work with are the arc length (L), the chord length (C), and the circular segment angle in degrees (A): I'm hoping someone can show me how to...
  5. knotted_pine

    Engineering Finding arc length of a pipe between two tanks

    I can't seem to find the arclength between A and B. I tried using L = integral (0.6 to 0.4) of sqrt (1+ (dz/dx)^2) to no avail. Would it be roughly similar to 400 km (the length from A to B) since the change in elevation could be considered negligible? Furthermore, how might I go about...
  6. O

    Arc length of vector function - the integral seems impossible

    The vector equation is ## v(x)=(e^x cos(2x), e^x sin(2x), e^x) ## I know the arc-length formula is ## S=\int_a^b \|v(x)\| \,dx ## I found the derivative from a previous question dealing with this same function, but the when I plug it into the arc-length function I get an integral that I've...
  7. T

    Multivariable Arc Length Problem: Weird Form with Parameterization

    Problem: See Attachment. Parts (a) & (b) are clear, but my confusion arises in (c)-- I feel like there is a much simpler form. While technically my answer is correct, there must be something I'm missing. I parameterized the curve C=(t, e^2t, e^2t) and got c'(t)=(1,2e^2t,2e^2t), which should...
  8. Adrian555

    A Geodesics of the 2-sphere in terms of the arc length

    I'm trying to evaluate the arc length between two points on a 2-sphere. The geodesic equation of a 2-sphere is: $$\cot(\theta)=\sqrt{\frac{1-K^2}{K^2}}\cdot \sin(\phi-\phi_{0})$$ According to this article:http://vixra.org/pdf/1404.0016v1.pdfthe arc length parameterization of the 2-sphere...
  9. Saracen Rue

    I How to find the maximum arc length of this equation?

    After seeing a discussion about graphs of the relationship ##x^x + y^y = r^r##, it got me interested in attempting to see what the graphical appearance of ##{^{\infty}x}+{^{\infty}y}={^{\infty}r}## would look like. The first step I did was use the relationship of...
  10. M

    Helicoidal movement: acceleration vector, arc length, radius of curvature

    I have tried to solve it and I would like a confirmation, correction or if something else is suggested... :) Helicoidal movement
  11. R

    I Arc diameter as a function of arc length and chord length

    I'm trying to determine if a certain bicycle tire size will fit my bike, and that determination is based on the inflated diameter (or width) of the tire. As such, I'm trying to come up with a formula that will give me the diameter of a bicycle tire as a function of the tire's carcass width and...
  12. T

    MHB Finding Chord length given arc length and arc height?

    I am in the process of making a tent and need to get two curved surfaces to meet. I need to find the length of a Chord of a circle given that I have the Arc length and Arc Height (that's all), no radius or anything else. I suspect that I will need a radius to find this. Or am I missing a...
  13. Zack K

    Trapezoidal rule to estimate arc length error

    I got the first part of it down, $$L=\int_1^5 \sqrt{1+(\frac{1}{x^2})}dx$$ I just want to know if it's right to make your ##f(x)=\sqrt{1+\frac{1}{x^2}}## then compute it's second derivative and find it's max value, for the trapezoidal error formula.
  14. E

    MHB Find radius, circumference, area and arc length in circle

    So I've been able to solve A.12 inch B.24 inch C.75.3982 D.452.389 Cant solve e. Arc length can anyone please explain the formula. Thank you
  15. opus

    I Looking for intuition on differentials and arc length

    Please see the attached image. It's a part of a study guide for my final, but I didn't put it in the homework section because I already got the answer, I just don't know what it means. The question has to do with the differential of an arc length. I made some drawings to see if I could make some...
  16. nomadreid

    I Arc length constant, despite varying the period via varying amplitude

    This is not a school problem, just my own mucking about, but since it has the form of a problem, I am willing to shift it to the "homework problems" rubric. If there is a theoretical string (no thickness, etc.) that is non-stretchable tied to two endpoints and is long enough to be able to form a...
  17. C

    MHB Secant and Tangent Angles in Circles, finding an arc length.

    I got 138.17, but that isn't correct. I don't know how to do it, since the only way I thought, gave me the wrong answer. Can anyone help?
  18. C

    Using dy/dx to find arc length of a parametric equation

    Homework Statement I have attached a picture of the problem in the attachments I need help on the last section, (part d) Homework Equations (1)##∫√( (dx/dt)^2+(dy/dt)^2)dt## (2)##∫√( 1+(dy/dx)^2)dx##[/B] The Attempt at a Solution In order to get the answer we just need to find the...
  19. H

    Calculating Arc Length and Area of Cylinder Side

    Homework Statement Let r be a positive constant. [/B] Consider the cylinder x2 + y2 ≤ r2, and let C be the part of the cylinder that satisfies 0 ≤ z ≤ y. (3) Let a be the length of the arc along the base circle of C from the point (r, 0, 0) to the point (r cos θ, r sin θ, 0) (0 ≤ θ ≤ π). Let...
  20. M

    MHB Find s, the arc length of a circle

    The question is asking me to find s, the arc length of a circle. How do I convert my answer in radians to the book's answer 7.54 meters? Theta = 4pi/5 radius = 3 meters See picture.
  21. I

    Arc Length of Parabola & Square Root Function

    Homework Statement Consider the curves: y = x^2 from 1/2 to 2 and y = \sqrt{x} from 1/4 to 4. a. Explain why the lengths should be equal. b. Set up integrals (with respect to x) that give the arc lengths of the curve segments. Use a substitution to show that one integral can be...
  22. R

    Help me find the arc length of a parametric equation....

    1. The problem statement, all variables and given/k nown data x = (sin(t))^2 y = (cos(t))^2 t goes from 0 to 3 pi Homework Equations ∫ \sqrt{ {(dx/dt)}^2 + {(dy/dt)}^2 } dt The Attempt at a Solution ∫ \sqrt{ {(2sin(t)cost)}^2 + {(-2cos(t)sin(t))}^2 } dt ∫ \sqrt{ 4(sin(t)cos(t))^2 +...
  23. R

    I Arc Length Parameterization for Unit Circle: Cos(s) & Sin(s)

    (cos(s), sin(s)) gives an arc-length parameterization of the unit circle so that the speed is constantly 1, but the second derivative doesn't give zero acceleration which should be the case with constant speed?
  24. M

    Show Regular Homotopy thru curve and its arclength parameters

    Homework Statement Let γ be a regular closed curve in Rn. Show that there is a regular homotopy Γ through closed curves with Γ(−, 0) = γ and Γ(−, 1) an arclength parametrization of γ Homework EquationsThe Attempt at a Solution Hey guys, I just posted another question about homotopy but often...
  25. M

    Show existence of arc-length parameterized period p'....

    Homework Statement Let γ : R → Rn be a regular (smooth) closed curve with period p. Show that there exist an orientation preserving diffeomorphism ϕ: R → R, a number p' ∈ R such that ϕ(s + p') = ϕ(s) + p and γ' = γ ◦ ϕ is an arclength parametrized closed curve with period p' Homework...
  26. S

    Finding length of an arc produced by projectile

    So I decided to try deriving a general formula for fun. Being a high school student, the calculus got scary very fast. At this point, I'm just curious as to what the best approach to this might be. The approach I used was finding y as a function of x and then inputting it into the arc length...
  27. M

    Invariance of length of curve under Euclidean Motion

    Homework Statement Show that the length of a curve γ in ℝn is invariant under euclidean motions. I.e., show that L[Aγ] = L[γ] for Ax = Rx + a Homework Equations The length of a curve is given by the arc-length formula: s(t) = ∫γ'(t)dt from t0 to tThe Attempt at a Solution I would imagine I...
  28. F

    MHB Arc length & similar questions

    Hello! I really don't understand this concept, and I have an example problem that I am working on that I just CANNOT figure out! Any help? Thanks so much in advance! A group of people get on a pirate ship ride at the fair. This ride is a swinging pendulum with a maximum swing angle of 65 degrees...
  29. bananabandana

    General proof of Arc Length For Parametrised Coodrdinates

    Homework Statement Prove that, given a metric ##g_{ij}## such that ##ds^{2}=g_{ij}dx^{i}dx^{j}##, where ##x^{r} = x^{r}(\lambda)## , we have the following result for the arc length: $$ L(p,q) = \int_{p}^{q} ds = \sqrt{ g_{ij} \frac{dx^{i}}{d \lambda} \frac{ dx^{j}}{d \lambda} } d \lambda $$...
  30. S

    I What is the purpose of Arc-Length Parameterization?

    My teacher just briefly introduced arc length parameterization and went on to frenet serret frames, without any explanation or motivation. What is the purpose of arc length parameterization? What role does it play in TNB? What is the purpose of TNB frames anyways?
  31. S

    Arc-Length Parameterization

    Homework Statement Find the arc-length parameterization for r(t)=\left< { e }^{ 2t },{ e }^{ -2t },2\sqrt { 2 } t \right> ,t\ge 0 Homework Equations s(t)=\int { \left| \dot { r } (t) \right| dt } The Attempt at a Solution \dot { r } (t)=\left< { 2e }^{ 2t },-2{ e }^{ -2t },2\sqrt { 2 }...
  32. J

    I Decomposing the arc length of a circular arc segment

    A particle travels along a circular arc segment centered at the origin of the Cartesian plane with radius R, a start angle θ1 and an end angle θ2 (with θ2 ≥ θ1 and Δθ = θ2 - θ2 ≤ 2π). The total distance traveled is equal to the arc length of the segment: L = R(Δθ). I would like to find the...
  33. LLT71

    B Why don't we use arc length formula to calculate wavelength?

    can you please explain me why don't we use arc length formula to calculate wavelength? seems a bit confusing...
  34. G

    Arc length in polar coordinates

    I know that an arc length in polar coordinates can be computed by integrating $$\int ds$$ using the formula ##ds=\sqrt{\rho^2 + \frac{dr}{d\theta}^2}d\theta##. But, seeing that ##s=\rho\theta## and ##ds = \rho d\theta##, why is it wrong to calculate arc lengths with this expression for ##ds##?
  35. 24forChromium

    Area of a sector is the integral of arc length?

    Area_sector = 0.5 (radius)^2 * angle Arc length= radius * angle Can it be said and proven that the area of a sector is the integral of the arc length? What would that even mean?
  36. G

    MHB Yes, your solution is correct!

    Consider the segment of the curve $y = \cosh(ax)/a$ between $x = −l$ and $l$. Here $a$ and $l$ are positive constants. Find an explicit expression for the length of this curve segment in terms of $a$ and $l$, as well as its limit for $a \to 0$. What I had done: Using the formula $\displaystyle...
  37. NicolasPan

    Comp Sci Error while calculating arc length of curve in Fortran

    The program must calculate the length of the curve of ƒ=3.1*x^2-5.3/x between x=1/2 and x=3/2.The legth should be calculated as the sum of n line segments starting with n=1 and ending with n=20. I really can't find why the result I'm getting is wrong.Thanks in advance I am giving you the code...
  38. adi adi

    Arc Length Circle Quadrant 1: Solve ∫√(1+(dy/dx)2)dx

    Homework Statement find the arc length of a circle in the first quadrant with an equation x2 + y2 = a2 Homework Equations arc length = ∫ √(1 + (dy/dx)2) dx The Attempt at a Solution i got stuck on how to solve the integral
  39. Dethrone

    MHB Arc Length Formula: Understanding the Proof

    I'm reading about Line Integrals, so I thought I'd review the proof for the arc length formula. However, there's something I don't quite understand about the proof that I either overlooked or understood before. From what I see, the arc length formula holds because the MVT guarantees that there...
  40. R

    Arc Length. Simplify expression

    Homework Statement I'm trying to find Arc Length of F(x) = (e^x + e^-x)/2 0< x < 2]Homework Equations L = integrate sqrt ( 1 + (dy/dx))^2)The Attempt at a Solution (dy/dx)^2 + 1 = 1/4e^2x + 1/4e^-2x + 1/2] I don't know how to take the square root of the above function so I can be able...
  41. P

    Finding Arc Length of a Curve: Using ##dx## and ##dy##

    The arc length of any curve defined by ##y = f(x)## is found as follows: $$ds = \sqrt{dx^2 + dy^2}$$ $$ds = \sqrt{dx^2(1 + {\frac{dy}{dx}}^2)}$$ $$ds = \sqrt{dx^2} \sqrt{1 + [f'(x)]^2}$$ $$ds = \sqrt{1 + [f'(x)]^2} dx$$ Isn't ##\sqrt{dx^2}## equal to ##|dx|##, and not ##dx##?
  42. M

    Solve Geometry Q w/ L and c: Find R in Terms of L and c

    Homework Statement In the attached drawing, find R in terms of L and c. Also, at the bottom of the picture I wrote something wrong. I said c, which equals 0.5, is the arc-length of each semi-circle, but I really meant to say each quarter circle. My bad. I'm not given a number for L so that can...
  43. RJLiberator

    Tricky integral from calc 3 Arc Length question

    Homework Statement Find the arc length of: r(t)=<e^t, e^(-t),sqrt(2)*t> from 0 to ln(2) Homework Equations L=integral from a to b of the magnitude of r'(t) The Attempt at a Solution Okay, this was an Exam question, the one exam question that I could not get on our Calc 3 exam. This breaks...
  44. RJLiberator

    Arc Length in Three Dimensions Question

    Homework Statement Find the arc length of r(t)= <tsin(t), tcost(t), 3t> from 0 to t to 2pi (inclusive) Homework Equations Integral from 0 to 2pi of the magnitude of r'(t) dt The Attempt at a Solution 1. Must find the derivative of the function. Using the product rule a few times, the...
  45. C

    Arc length of a stadium billiard

    I've been trying to figure out the most straightforward way of doing this for a while, and would like to get some advice on new approaches, as the one I was using didn't work out at all. So here it is: The stadium billiard is defined as two semicircles joined by two tangent lines, as shown in...
  46. Calpalned

    Deriving the formula for arc length of a polar function

    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...
  47. J

    Find the arc length (using hyperbolic trig)

    1. The problem statement, all variables and given/known Find the length of the curve $$y=ln(x),\frac{1}{2}<=x<=2$$ Homework Equations Using hyperbolic trig isn't necessary, but it's how my text (Serge Lang's A First Course in Calculus) approaches most square roots, and as a result, it's what...
  48. A

    MHB Arc length question

    The length of the minor arc of a circle is 10cm, while the area of the sector AOB is 150cm2. a) Form two equations involving r and θ, where θ is measured in radians. b) Solve these equations simultaneously to find r and θ. Help to solve? Cant understand the question very well. I think the...
  49. A

    Precalc Trig Arc Length Question

    Homework Statement http://education.Alberta.ca/media/9451970/07_math30-1_released_2014-15.pdf question #7 Homework Equations a=theta*r The Attempt at a Solution I did a=(5pi/6)*20 but the answer is not A