# Questions tagged [arc-length]

For questions about/on finding the arc length of a curve/parametrized curve

757 questions
Filter by
Sorted by
Tagged with
6k views

### Graphs for which a calculus student can reasonably compute the arclength

Given a differentiable real-valued function $f$, the arclength of its graph on $[a,b]$ is given by $$\int_a^b\sqrt{1+\left(f'(x)\right)^2}\,\mathrm{d}x$$ For many choices of $f$ this can be a ...
• 19k
2k views

### Can you define arc length using a piece of string?

In calculus, how we calculate the arc length of a curve is by approximating the curve with a series of line segments, and then we take the limit as the number of line segments goes to infinity. This ...
• 10.3k
64k views

• 12.4k
902 views

### An amazing property of the Catenary

I discovered that if we want an arc of catenary in the interval $[a,b]$ we solve $$\int_a^b \sqrt{\cosh '(x)^2+1} \, dx=\int_a^b \cosh x \, dx$$ which means that the "result" of the length ...
• 26.4k
1k views

### Is it possible to find the coordinates of a point on the circumference of a circle, without using trigonometric functions?

I don't have a particularly good reason to want to do this, and I'm just asking out of curiosity. I am looking for the coordinates of point $\pmb B$, a point on the circumference of a circle. If I ...
• 113
822 views

### Approximating the length of a circular arc using geometrical construction. How does it work?

I was going through my Engineering Drawing textbook and came upon this topic. Using only a compass and a straightedge, one can supposedly approximate the length of a given circular arc by following ...
981 views

### Integral inequality of length of curve

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuously differentiable function. Prove that for any $a.b\in \mathbb{R}$ $$\left (\int_a^b\sqrt{1+(f'(x))^2}\,dx\right)^2\ge (a-b)^2+(f(b)-f(a))^2$$. ...
• 1,246
1k views

### Given a circle of radius r, and two points ('X' and 'Z') on that circle, can some circumcircular arc "XYZ" be constructed of length r?

I am strictly an amateur, not a professional mathematician or some such. This question occurred to me while considering the fact that an angle of 1 radian centered on the center of a circle will ...
• 193
1k views

103 views

### How do I find the arc length of $y=1-e^{-x}$ from $0 \leq x \leq 2$?

I can set up the integral $\int_{0}^{2} \sqrt{1+e^{-2x}}\,dx$ by taking the derivative of y and by using the arc length formula. I'm really stuck on how to evaluate this integral. I've tried to follow ...
• 101
Question: Use Definition 3.2 to prove Theorem 3.4. Definition 3.2 “The signed curvature $k(s)$ of a plane curve $\alpha: I \rightarrow \mathbb{R^2}, \alpha(u)=(x(u),y(u))$ is defined by $t’(s)=k(s)... • 415 5 votes 1 answer 635 views ### Can the strict triangle inequality hold in a length space? Let$(M, d)$be a metric space where (0) for relevance: set$M$contains at least three distinct elements and distance function$d : M \times M \to \mathbb R$explicitly satisfies (1) non-... • 491 5 votes 1 answer 94 views ###$\ell(f)>\int_{a}^{b}||f'(t)||dt$We know that if$f:[a,b] \rightarrow \mathbb{R}^{n}$is a$C^1$path then $$\ell(f)=\int_{a}^{b}\|f'(t)\|dt.$$ Moreover, in the proof of this result we use explicitly the continuity of$f'.$I'm ... • 2,379 5 votes 1 answer 120 views ### Prove convergence of formula for length of parameterized curve In class, our professor was discussing the arc length of a parameterized curve$x=f(t), y= g(t)$. In his derivation, he reached the sum - $$\sum_{i=1}^n L_i = \sum_{i=1}^n \sqrt{(\Delta x_i)^2 + (\... • 3,604 5 votes 2 answers 617 views ### Interpolation method that gives the least arc lenght of the curve. this question may be a bit sloppy on my part but I will make it anyway, I recently have been fascinated by the idea that the surface of a soap bubble film restricted to a boundary will be so as to ... • 167 5 votes 1 answer 2k views ### How to determine X,Y position from point P based on time, velocity and rate of turn? We need to figure out an estimated position along an arc using a set of known variables. The starting position on the arc would be called point P, here are the known/measured variables: P = initial ... 5 votes 2 answers 2k views ### Lower semicontinuity of length of graph: L(g)\le\liminf_{n\to\infty}L(f_n) I am interested in the following claim:\newcommand{\intrv}[2]{[#1,#2]}\newcommand{\limti}[1]{\lim\limits_{#1\to\infty}} Let g,f_1,f_2,\dots be continuous functions on \intrv ab such that g=\... • 54.2k 5 votes 1 answer 359 views ### "Unrolling" a 3d wedge I have a parametric equation that describes a particular intersection in 3d space. I'd like to flatten this by deforming the object without stretching its length, as if this were the outline of a ... • 1,252 5 votes 0 answers 220 views ### Find point of n-sphere after moving on geodesic by given angle Let \mathbb{S}^n=\{\mathbf{x}\in\mathbb{R}^{n+1}\colon\lVert\mathbf{x}\rVert^2=1\} be the unit n-Sphere and \mathbf{x}\in\mathbb{S}^n. Given a vector \mathbf{a}\in\mathbb{R}^{n+1} and an angle \... • 3,076 5 votes 0 answers 303 views ### Ideas for parameterizing this curve in the complex plane and calculating its length by (numerical) contour integration? Let Z (t) be the Hardy Z function. Then define $$Y (t) = \tanh (\ln (1 + Z (t)^2))$$ Let us define the length L_n of the curve which intersects the n-th Zero of Z (... • 1 5 votes 0 answers 88 views ### Supremum of arc lengths of graphs of power towers Consider the set of all functions of one variable x\in[0,1] that can be constructed from any number of instances of that variable using parentheses and exponentiation only:$$x,\;x^x,\,x^{x^x},\;\... • 47.3k 4 votes 2 answers 174 views ### Calculus 2 test question that even the professor could not solve (Parametric Arc length) everyone got this question wrong from my calculus 3 midterm. " find the arc length of the curve on the given interval:$x=\sqrt(t), \space y=8t-6,\space on\space 0 \leq t \leq 3$" I set the ... 4 votes 1 answer 194 views ### Calculating the arc length of a curve... Which formula? Let$S$be a surface parameterized by variables$u,v$and$\alpha(t)=(u(t),v(t))$be a curve on the surface. I am of the understanding that we can find the arc length of$\alpha$by integrating it's ... 4 votes 3 answers 8k views ### Finding the arc length of the parabola$y=x^2 \; from \; (0,0)\;to\;(1,1)\$
As the title says, I need to find the arc length of that. This is what I have so far (I'm mostly stuck on the integration part): $${dy\over dx}=2x \Rightarrow L=\int_0^1 \sqrt{1+(2x)^2}dx$$ Substitute ...