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63 votes
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Graphs for which a calculus student can reasonably compute the arclength

Ferdinands, in his short note "Finding Curves with Computable Arc Length", also comments on the difficulty of coming up with suitable examples of curves with easily-computable arclengths. In ...
J. M. ain't a mathematician's user avatar
36 votes
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Is the arc length always irrational between two rational points?

Obviously, a straight line between two rational points can have rational length $-$ just take $(0,0)$ and $(1,0)$ as your rational points. But a curved line can also have rational length. Consider ...
TonyK's user avatar
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35 votes

Is the arc length always irrational between two rational points?

An example of a curve with rational arc lengths between at least some pairs of rational points is a cardioid. Down to scaling and rotation, a cardioid may be rendered in polar coordinates by the ...
Oscar Lanzi's user avatar
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29 votes

Graphs for which a calculus student can reasonably compute the arclength

I will dissent here (as often) and say: DON'T. The problem here is looking at everything as something that needs to be computed, "solved", or otherwise manipulated into some set, pat form. ...
The_Sympathizer's user avatar
28 votes

Graphs for which a calculus student can reasonably compute the arclength

Another example: you can get $$ \sqrt{1 + [f'(x)]^2} = ax + \frac 1{2ax} $$ by taking $f(x) = \frac 12 a x^2 - \frac 1{4a} \ln(x)$ for any constant $a$. A possibly helpful way of reframing the ...
Ben Grossmann's user avatar
23 votes

Is the arc length always irrational between two rational points?

So, my question is that do all curved path have irrational lengths? Of course not. A circle with radius $\frac{1}{2\pi}$ is a curved path and has length $1$ which is a rational number. If you put the ...
5xum's user avatar
  • 124k
23 votes

Graphs for which a calculus student can reasonably compute the arclength

This example $$ y = a\cosh \frac{x}{a} $$ is quite simple for computations.
Virtuoz's user avatar
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22 votes
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Conjectured connection between $e$ and $\pi$ in a semidisk.

$\newcommand{\eps}{\varepsilon}$ The answer is in fact not $\pi/2$ but $$\frac{3}{(2\pi)^{1/3}}\prod_{k=1}^\infty\left[1-\big((k+1)^{1/3}-k^{1/3}\big)^3\right]\approx 1.56813\ldots.$$ This product ...
Carl Schildkraut's user avatar
19 votes

Graphs for which a calculus student can reasonably compute the arclength

You can try $f(x)=\dfrac{\sqrt{a^2e^{2ax}-1}-\tan^{-1}\sqrt{a^2e^{2ax}-1}}{a}$, which has arc-length $e^{ax}-1$ and isn't too hard to work with as long as you remember $\frac{\mathrm{d}}{\mathrm{d}x}\...
Jam's user avatar
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18 votes
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Why $\pi r$ is not equal to $2r$?

You're making an assumption here: if the limit (in some sense) of a set of curves is some limit curve (the diameter in your case), then the limit of the LENGTHs of the curves must exist and equal the ...
John Hughes's user avatar
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18 votes
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Is this "derivation" of the path length formula actually correct?

What he did is correct, though the reasons for doing so seem to be glossed over. I'll give you a rigorus version of what he did. Let $\Delta x > 0$, represent the length of some horizontal line ...
d4rk_1nf1n1ty's user avatar
18 votes
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Integral inequality of length of curve

Notice that the function $y \mapsto \sqrt{1+y^2}$ is strictly convex. So by the Jensen's inequality, $$ \frac{1}{b-a} \int_{a}^{b} \sqrt{1 + f'(x)^2} \, \mathrm{d}x \geq \sqrt{1 + \left(\frac{1}{b-a}\...
Sangchul Lee's user avatar
17 votes

Why do I get two different answers when solving for arclength?

Your second formula applies when you see $y$ as a function of $x$; you don't say how you found $dy/dx$. Playing a bit loose with differentials, we have $$ \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{...
Martin Argerami's user avatar
15 votes

Is this "derivation" of the path length formula actually correct?

This is what helped me stay afloat during general relativity classes; it might help you too. First note, that this is always in the context of some (given or arbitrary) path $S$. Say your path is ...
Arthur's user avatar
  • 201k
15 votes

Integral inequality of length of curve

Note that for every complex valued integrable function $\phi :[a,b]\to \Bbb C$, it holds that $$ \left|\int_a^b \phi(x)\ dx\right|\le \int_a^b|\phi(x)|\ dx. $$ Let $\phi(x)=1+if'(x)$. Then we can see ...
Myunghyun Song's user avatar
15 votes

Is the arc length always irrational between two rational points?

Consider the two points $(-\frac12,0)$ and $(\frac12,0)$. For any real value of $y_0$, we can draw a circular arc between these two points which is centered at $(0,y_0)$ and which lies entirely in ...
Michael Seifert's user avatar
13 votes

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

The trig functions provide a dictionary between the arc measurement of an arbitrary$^\dagger$ point on a circle (or similarity class of a right triangle described by an angle) and the rectangular ...
Sammy Black's user avatar
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13 votes
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What does a function of its own arc length look like?

No such function exists. (Well, but see below ...) A starting point which you've essentially gotten to already is to ask whether there is a continuously differentiable (or even nicer) function $f$ ...
Noah Schweber's user avatar
11 votes
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Is it possible to find the coordinates of a point on the circumference of a circle, without using trigonometric functions?

The answer to your title question is yes (stereographic projection). However, given the information you have, no, not really, because one of the pieces of information you're given is the length of the ...
peek-a-boo's user avatar
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9 votes

Curve with longest arclength between two points

tl; dr: Despite the constraint, there are smooth paths of arbitrarily long length. Call the points $p_{0} = (0, 0)$ and $p_{1} = (1, 0)$, i.e., assume $d = 1$. Fix a positive integer $n$ and consider ...
Andrew D. Hwang's user avatar
9 votes
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Length of a super-circle

We start with $$L(n)=4n\int_0^{\pi/2}\sqrt{\cos^{2n-2}\theta\sin^2\theta+\sin^{2n-2}\theta\cos^2\theta}\,d\theta$$ and then substitute $t=\sin\theta$, then $u=t^2$ to get $$L(n)=2n\int_0^1\sqrt{u^{n-2}...
Parcly Taxel's user avatar
8 votes
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Why isn't the arc length of $\cos x$ equal to $\pm \sin x$?

$$(-\sin x)^2 \ne -(\sin x)^2$$
Kenny Lau's user avatar
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7 votes
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How do I calculate the chord length I need to produce this Japanese sun pattern?

If you want the nested arcs to be concentric, you can choose centers so that they form a triangular lattice: $\hspace{11.1em}$ Then the arc length is $\frac{1}{3}$ of the full circle, which is ...
Sangchul Lee's user avatar
7 votes
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Finding the arc length of the parabola $y=x^2 \; from \; (0,0)\;to\;(1,1)$

Let $2x = \tan\theta$ instead. Then, the integral becomes $\displaystyle \int_0^{\arctan 2} \sqrt{1+\tan^2 \theta} \cdot \dfrac14\sec^2\theta \ \mathrm d\theta$ which is equal to $\displaystyle \...
Kenny Lau's user avatar
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7 votes
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Arc Length Integral of $x^x$ from 0 to 1 in closed form.

One trivial solution to this problem that was just noticed was using the definition of the integral with one variable and the other estimation techniques likely give the same summation such as the ...
Тyma Gaidash's user avatar
7 votes
Accepted

Why is arc length independent of parametrization?

Assuming $A = [c,d]$ and $B=[e,f]$, there is a change of parameter diffeomorphism $h : [c,d] \to [e,f]$ such that $x_1(t)=x_2(h(t))$ and $y_2(t)=y_2(h(t))$, $t \in [c,d]$. Letting $s = h(t) \in [e,f]$ ...
Lee Mosher's user avatar
  • 123k
7 votes
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Function equal to its own arclength.

Suppose there was such a function $f$. Fix an $x > 0$. Since the arclength of $f$ between $0$ and $x$ is at least the length of the straight line between $(0,f(0)) = (0,0)$ and $(x,f(x))$, we have $...
JimmyK4542's user avatar
  • 54.4k
6 votes

Evaluating the arc length integral $\int\sqrt{1+\frac{x^4-8x^2+16}{16x^2}} dx$

Hint $$1 + \frac{x^4 - 8 x^2 + 16}{16 x^2} = \frac{16 x^2}{16 x^2} + \frac{x^4 - 8 x^2 + 16}{16 x^2} =\frac{x^4 + 8 x^2 + 16}{16 x^2}$$ Can you write the rightmost expression as a square of another ...
Travis Willse's user avatar
6 votes
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Identical Geodesics implies scalar multiple of metric?

No. Consider as your model set $M$ a tripod. (A tripod is a graph with one vertex of degree three and three vertices of degree one attached to it.) For your different length metrics, just assign the ...
Anon's user avatar
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6 votes
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Why arc-length parametrized curves has unit tangent vector?

Yes, provided your curve has nonzero tangent vector at all points. Suppose your curve is $\alpha: [a, b] \to \Bbb R^2$. For $t \in [a, b]$, define $$ q(t) = \int_a^t \| a'(s) \| ds. $$ You can see ...
John Hughes's user avatar
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