# Questions tagged [arc-length]

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

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### sequence of abscissas x corresponding to equispaced points on any function

I am writing a program where I need to generate the sequence of points $x_1$ , $x_2$, $x_3$,$...$ such that the corresponding ($x_1$, $y_1$), ($x_2$, $y_2$), ($x_3$, $y_3$), $...$ on a generic ...
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### How do I determine the plastic section modulus Z of an arc? [closed]

I am trying to find a simplified formula for determining the plastic modulus, Z, of an arc of a circle. In my case, I have a circle that has a diameter of 20" and I am removing an arc length of 6&...
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### Arc Length of the parabola

Can You please tell me how to derive this formula? $$L=\frac12\sqrt{b^2+16a^2} + \frac{b^2}8a \ln \left(\frac{4a+\sqrt{b^2+16a^2}}b\right)$$
1 vote
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### Can't spot my error in calculating 3D Parametric Arc Length

We're asked to find a function s(t), for the arc length of a curve centered at point t=0, as a function of t. The function is as follows... $\gamma (t) =e^t i + \sqrt{2} tj-e^{-t}k$ My work is as ...
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### How to solve these complicated SA and arc length integrals?

My partner and I are working on a project for our multivariable calculus class where we have to solve the integrals to find the arc length and surface area of our three piecewise functions. We've used ...
1 vote
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### Polar curves with specified length function

Given a probability density function $f(x)$ with support $[0,2\pi]$, I'm interested in constructing a "roulette" in which the "winning angles" follow that distribution. The shape ...
1 vote
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### Arc-length change of variable in the inviscid Burgers equation.

Being quite new to the world of PDEs, I would like your help regarding a specific change of variable. Namely, I consider the inviscid Burgers equation : \begin{equation} u_t+uu_x=0 \end{equation} And ...
1 vote
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### how to measure the arc length?

We're told to measure angles in radians, θ = arc length/radius. Therefore, 1 radian occurs when the radius of the circle is equal to the arc length subtending the angle you're looking to measure (θ). ...
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### On the curve $y=\frac{\sin (\pi x)}{x^p},x>0$, for what values of $p$ does the product of all the arc lengths between neighboring roots exist?

Consider the curve $y=\frac{\sin (\pi x)}{x^p}, x>0$, shown here with $p=0.75$. It occurred to me that if $p$ is large enough, then the curve flattens quickly, so the arc lengths between ...
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### Deriving the formula for length of function curve.

In my textbook the first step in deriving the length of a function curve is to the the sum $\lim_{n\to \infty}\sum_{i=0}^{n}\sqrt{(x_{i}-x_{i-1})^{2}+(f(x_{i})-f(x_{i-1}))^{2}}$ Now my text continues ...
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### Calculate length of involute without calculus

Is it possible to calculate the length of an involute of a circle without using calculus, and how? Say we’re interested in the involute length for $\theta$ between $0$ and $2 \pi$.
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### Arc length for continuous implicit differentiable functions

We have a continuous differentiable function defined as $$F(x,y)=0$$And I am looking for a formula for its arc length between $x$ values $a$ and $b$. Doing a quick search, I could only find formulae ...
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### A person's quickest path between any two points on perimeter of elliptical lake never involves both swimming and running. Find maximum eccentricity.

I made up this question. A police officer's job is to patrol the perimeter of an elliptical lake. They have a constant (unknown) swimming speed and a constant (unknown) running speed. Their quickest ...
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### Two Distinct Arc Length Parameterisations

$f$ and $g$ are two arc length parameterisations of a regular simple curve $C$, then prove either $f(s) = g(e+s)$ or $f(s) = g(e-s)\; \forall s$ for some constant $e$. Intuitively I can understand as ...
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### Calculating the length of a rope hanging from one point

If you have a rope hanging from one point in the air, how can you calculate the length of it (without measuring it). I don’t really know anything that complicated about maths, but I’m curious if it is ...
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### Length space and continuity assumption on curves

I think the definition of length metric works without assuming curves are continuous. Let $X$ be a subset of $\mathbb{R}^n$ and $x, y\in X$. Def 1. For a function $f:[0, 1]\rightarrow X$, $L(f)$ is ...
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### Apostol, Vol I Ch. 14: When defining an arc length function $s(t)=\Lambda(a,t)$ why is it necessary to specify $s(a)=0$?

In Apostol's Calculus, Ch. 14 "Vector-Valued Functions" section 14.12, he derives the result that the function representing the arc length of a parametric curve is given by the integral of ...
1 vote
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