# What is Arclength: Definition and 28 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. ### 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...
2. ### University Physics: Force Problem (Baseball and Pitcher)

I've gone about getting the arclength S = 1.103m The formula for average force is F = m * (Vf – Vi) / t I know the mass and the initial velocity, but I don't know where arclength comes into play. I'm assuming Vf and T is referring to the moment that the ball leaves the pitchers hand, but I...
3. ### 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...
4. ### 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...
5. ### I Velocity with respect to arclength is a unit tangent vector?

Hi all, I have long had this unsolved question about arclength parameterization in my head and I just can't bend my head around it. I seem not to be able to understand why velocity with arclength as the parameter is automatically a unit tangent vector. My professor proved in class that s(s) =...
6. ### Arclength Problem; stuck on integration

Homework Statement Find the arclength of $$r(t) = <{t}^2/2,{t}^3/3>$$ from t=-1 to t=1 Homework Equations I have used this equation for arclength $$\int_{-1}^{1}|{r}'(t)|dt$$ The Attempt at a Solution After integrating (using u substitution) I have the solution...
7. ### Find arclength of curve; stuck trying to integrate radical

Homework Statement Find the arclength of the curve x = ⅓(y2+2)3/2 from y=0 to y=1. Homework Equations (Please forgive the crazy definite integral symbols - I'm taking a LaTex class tomorrow so hopefully I'll be able to communicate more clearly from then on..!) arclength of curve = ∫ab √ (1 +...
8. ### Finding the radius of a Proton's arc inside a square.

Homework Statement This is for a practice question on an exam: I am able to finish the problem, if I could figure out how to find the radius of this arc the proton makes. Homework Equations I have nothing. The Attempt at a Solution I have tried arc length equations and just integrating the...
9. ### Arclength parameter for a path with exponential and trigonometric functions

Homework Statement Find the arclength parameter s=s(t) for the path x(t)=e^(at)cos(bt)i + e^(at)sin(bt)j + e^(at)k Homework Equations s(t)=\int_{a}^{t}\left | \mathbf{x}(\tau ) \right |d\tau The Attempt at a Solution I took the derivative and squared it, arriving at the equation (-be\sin...
10. ### How can I implement a Pseudo Arclength Continuation method for my system?

I'm struggling to implement a pseudo arclength continuation method for my system. Here is what I have so far. I am trying solve the system of equations F(x, \lambda) = 0 but if I parameterise only by using lambda, I can't get around turning points, so I paramterise by "arclength" s and attempt...
11. ### Calculate Arclength of Parametric Curve

Homework Statement Calculate the arclength of the curve given parametrically by ## x=2t^2, y=\frac 8 5 \sqrt 3t^ \frac 5 2, z=2t^3 ## for 0≤t≤2 Homework Equations ## S=∫ \sqrt(dx^2 + dy^2 + dz^2) ## The Attempt at a Solution 1. Found derivative of each and input into equation. ##...
12. ### Rewrite curve as arclength function

Homework Statement Consider the curve r = <cos(3t)e^(3t),sin(3t)e^(3t),e^(3t)> compute the arclength function s(t) with the initial point t = 0. Homework Equations s = integral |r'(t)|dt The Attempt at a Solution Okay so if you work all of this out it turns out it's not as bad as...
13. ### Arc Length of Parametric Curve: t = 0 to t = 1

Homework Statement Find the arc length of a curve given parametrically from t = 0 to t = 1. Curve given by x = 4t^2, y = 2t Homework Equations [I think] parametric arclength = integral from t = b to t = a of sqrt( (dx/dt)^2 + (dy/dt)^2)dt The Attempt at a Solution dx/dt =...
14. ### Arclength of Curve y=2ln(sin(x/2)) | Calculating Arclength Formula

Homework Statement Find the length of the curve y=2ln(sin\frac{1}{2}x), \frac{\pi}{3}\leq x\leq\pi Homework Equations The Attempt at a Solution Alright so I figured out the derivative of y is cot(1/2)x so I put it into the arclength formula to get: \int_{\frac{\pi}{3}}^{\pi}...
15. ### The arclength of a parametrized segment (integration).

Homework Statement For two points p and q in ℝ^n, use the formula (20.3) to check that the arclength of the parametrized segment from p to q is ||p - q||. Homework Equations Formula (20.3): A smooth parametrized path \gamma: [a, b]→ℝ^nis rectifiable, and its arclength l is given by l...
16. ### Arclength of polar cardiod problem

Homework Statement My textbook sets up the integral, but does not solve, claiming that it's "trivial to solve manually or by using a CAS". I put the integral into my TI-89, and sure enough, there is a solution, and that solution happens to be "8". However... Homework Equations The actual...
17. ### Arclength method non linear fea

i would like to understand the arc length and line search method against Newton raphson method how the stiffness matrix updation and the general procedure is done my question is with reference to non linear fea
18. ### Calculating the Arc Length of y=e^x from 0 to 1

Homework Statement Find the arc length of y=e^x, from [0,1]. Homework Equations The Attempt at a Solution s = \int_0^1 (1 + e^2^x)^(^1^/^2^)dx I let t = e^x, dt=e^xdx; therefore dt/t=dx s = \int_1^e \frac{(1+t^2)^(^1^/^2^)}{t}\right) dx Let t = tanT, dt = sec^2(T)dT (T...
19. ### Arclength Calculation Process for y=x^3/2, 0<x<5/9

this is a really easy question... but seems like i keep getting something wrong in my calculations... - - Find the arclength of y = x^3/2 0<x<5/9 when i do it using the formula; L = sqreroot (1+(dx/dy)^2) i keep getting one... can someone post the calculation process...
20. ### Arclength on a PseudoRiemann Manifold

Wikipedia gives a confusing definition of a path's length and I would like some clarity. Let M be a pseudo-Riemann manifold with metric g and let a and b be points in M.If y is a smooth function from R->M where y(0) = a and y(1) = b, then it's length is the integral \int_0^1\sqrt{\pm...
21. ### Computing The Arclength Function

Homework Statement Consider the curve r = (e^−2 t cos(3 t), e^−2 t sin(3 t), e^−2 t) . Compute the arclength function s(t) : (with initial point t=0 ). The Attempt at a Solution r'(t) = <-2e^-2t*cos(3t) + e^-2t*-3sin(3t), -2e^-2t*sin(3t) + e^-2t*cos(3t), -2e^-2t> Then what...
22. ### Arclength, Surface Area and Volume

I was just thinking: If \iint dS is the surface area of a level surface, S, and \iiint dV is the volume of an enclosed solid, V, shouldn't \int df be the arclength of a function f(x)? Lets say that our surface is given implicitly by \Phi For the surface area we get: \iint dS =...
23. ### Arclength Math Question: Find a Closed Form Expression for x(t) and y(t)

Homework Statement Given r(t) = x(t)\textbf{i} + y(t)\textbf{j} and \int_0^t \left\|\frac{dr}{dt}\right\| d\tau = t. find, if possible, a closed form expression for x(t) and y(t). Homework Equations The Attempt at a Solution I started by applying the fundamental theorem of calculus...
24. ### What is the formula for finding arclength when given an integral?

Homework Statement Find the arclength of the curve given by y= integral from -pi/2 to x of sqrt(cost)dt. X is restricted between -pi/2 and pi/2. Homework Equations L = Integral from a to b of sqrt((dy/dx)^2 + 1)dx L = Integral from a to b of sqrt((dy/dt)^2 + (dx/dt)^2)dt The...
25. ### Arclength in polar coordinates

I am working on a problem regarding arclength-which asks to find the arclength for r=2-2sinx (x=theta) I worked out the integral to the integral of the square root of 8-8sinx but i didnt know how to integrate from there--any help? Thanks -nate808
26. ### Minimizing arclength with a set of parameters

i am trying make a function who's integral is equal to 1 from 0 to 1 and goes threw points (0,0) and (1,0) with the minimal arc length, is tehre anyway to do this? i have tried a couple things, which iwll get me xome question, i have made a triangle, and i have made many trapazoids, the...
27. V

### Arclength in polar coordinates?

You know this should be simple but it's just not. A friend asked me this earlier and I was unable to disprove him. We're all aware of how one derives the area of a polar equation.. it's \pi r^2 \frac {\theta}{2 \pi} and make theta infinitely small and integrate. Why can't a similar process...
28. ### Reparametrizing a space curve in terms of arclength:

Hello folks! I'd be happy for any help you could give me. Thanks! :) First though I'd like to make clear that I want to solve this problem myself, even if something seems obvious to you and you feel there is no harm in making a certain logical leap - please take the time to consider whether...