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Questions tagged [arc-length]

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

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Does there exist an opposite to curve length in integral calculus, “radius length”?

Consider the formula for curve (arc-) length in integral calculus: $$\int_a^b\sqrt{1+\left(\frac{\partial y}{\partial x}\right)^2}dx$$ What would the geometrical opposite thing to measure be? Some ...
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1answer
54 views

Finding arc length of $f(x) = \sqrt{x+2}$ - is there an easier way?

Find an arc length of $f(x) = \sqrt{x+2}$ $\quad \land \quad x\in<0,2>$ $$f'(x) = \frac{1}{2\sqrt{x+2}}$$ $$\int_{0}^{2} \sqrt{1+[f'(x)]^2}dx = $$ $$\int_{0}^{2} \sqrt{1+\Bigg[\frac{1}{2\sqrt{...
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2answers
38 views

How to calculate $\int \sqrt{1+\frac{4x^2}{(1-2x)^2}}dx = $?

I want to find an arc length of y = $\ln{(1-x^2)}$ on the $x\in<0,\frac{1}{2}>$ interval? $y' = \frac{-2x}{1-x^2}$ So: $$\int \sqrt{1+\frac{4x^2}{(1-2x)^2}}dx = $$ $$= \int \sqrt{\frac{(1-2x)...
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1answer
61 views

Arc length of $x^3 \sqrt{9-x}$ on $[0,9]$

This is supposed to be part of a student's Calc 2 homework; however, this seems to be an extremely difficult integration, and I couldn't figure it out. Find the arc length of $x^3 \sqrt{9-x}$ on the ...
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1answer
24 views

How to Calculate Point on an Arc (Given: Center, Arc Endpoints, Direction, and Distance along the Arc)? [duplicate]

I need to calculate the grid coordinates for a point on an arc using the distance traveled along it. I'm given the endpoints, radius, center, clockwise direction, and distance. How can I calculate ...
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1answer
36 views

What is the “Full Cycle” of a Polar Curve?

In order to find the arc length or area etc of a polar curve, you must integrate from $\theta_1$ to $\theta_2$. However, I'm having trouble finding the values of $\theta_1$ and $\theta_2$. I know ...
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Distance between two points at same angle in trochoid curve

Anyone please help me to find out the distance in following case. Refer to the attached image. Consider an arbitrary point P on the circumference of a circle of radius r (mm). The point makes an ...
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1answer
80 views

The arc length of $y = x^2$

I'm trying to find the arc length of the curve $y = x^2$ between points $a$ and $b$, meaning a formula where I input $a$ and $b$ and get the length of the arc as an output. I know about the ...
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2answers
31 views

Arc length of quadratic curve

I would like to find the arc length of a curve from $a\le t\le b$, the curve is $t^2A+tB+C$ $$arcLength=\int_{a}^{b}\sqrt{(2At+B)^2+1}\,dt$$ I am having trouble getting rid of $t$ (the variable) (...
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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},\;\...
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4answers
485 views

Integral inequality of length of curve

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

About the definition of the length of a curve (keyword: rectifiable)

If the upper and lower integrals are equal, we say that $f$ is Riemann-integrable on $[a, b]$. We use $\sup$ and $\inf$ to define the area under a curve. We associate to each partition $P = \{...
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1answer
49 views

Proof of formula for the arc length

I've read the proof of formula for arc length . I wonder why the function has to have continuous derivative (According to the Stewart Calculus book). I mean in which part of the proof we used this ...
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1answer
29 views

How to compute acute triangles and their values from sections of a circle?

I am working with a sensor that outputs data in forms of radial beams. In order for it to be useful for my application I have to compute a value of one side of the acute triangles that can be seen in ...
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1answer
20 views

Arc Length: Surface of revolution

The given arc is $y = 1 - \dfrac {x^2}{4}$ about the $y$-axis from 0 to 2. Here is the farthest part I could ever go through. $$y’ = -\frac{x}{2}$$ $$[y’]^2 = \frac{x^2}{4}$$ So $$\int_0^2x\cdot\...
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2answers
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How to find arc length (exponential function)

I am sorry I don't know how to use the MathJax equations format properly. I have to find arc length of $ y=( e^{x/2} + e^{-x/2} ) $ over this interval [-2,2]. I found the derivative of y and ...
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1answer
36 views

I am confused as to what kind of identity was used to find the arc length. Any help I would appreciate. Thank you! [closed]

I don't just understand how they simplified whats inside the <> into just $\sqrt 2$ Shouldn't it look something like: $\sqrt{(\cos2x - \sin2x)^2 + 1}$ ??
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2answers
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I am confused about how $\sqrt{\sin^2 6t +\cos^2 6t + (6\ln(\cos t))^2}$ simplifies to $36 \sec t$

$$\int (\sin^2 6t +\cos^2 6t + (6\ln(\cos t))^2)^{1/2} dt $$ $$\int (36 (\sin^2 6t + \cos^2 6t + \tan^2 t) )^{1/2} dt $$ $$\int 36 \sec t\,dt $$ the part that I do not get is when $6\ln(\cos t))^2$ ...
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1answer
41 views

Try to find the ratio of two arcs

O1 is the center of the big circle; O2 is the center of the small one. Line(O2-P) is vertical to line(A-C). Line(PQ):line(PB)=2:7 What's the ratio of arc(AC) and arc(CB)?
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Computing the arc length of the graph $y=\sqrt{x-x^2}+\arcsin(\sqrt{x})$

Computing the arc length of the graph $y=\sqrt{x-x^2}+\arcsin(\sqrt{x})$ Is this done right? I found the interval of this function which is $[0,1]$ I know the arc length formula is $$ L=\int_a^b\...
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Calculus Integration of Arc Length

Let $f(x)$ be a twice differentiable function over $[a,b]$ with arc length $L$. Show that there exists a value $c \in [a,b]$ such that the angle $\theta$ between the tangent line and the horizontal ...
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1answer
28 views

How are the steps to the solution for Arc - Length obtained?

Can someone please help me follow and understand the steps of the solution marked with $(*)$ and $(@)$? Why is the dot product used and computed with the unit vector. How does this equal the integral? ...
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1answer
12 views

Explanation on the steps of this total arc-length solution

Can someone please explain to me what rules have been used to calculate ${\bf{\dot{x}}}$ and $|{\bf{\dot{x}}}|$ in the definition for total arc-length of this problem. I've tried to calculate this by ...
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1answer
62 views

Tough integral from a falling clock. $\int_0^{2\pi} \sqrt{g^2t^2 + 2rgt\sin(t) + r^2} {\rm d}t$

A clock is under free fall for $60$ seconds and its second hand makes exactly one revolution during that period of time. It begins at rest with its second hand facing upwards. Given that the second ...
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0answers
19 views

arc length of a curtate cycloid

is there an equation that expresses the arc length of a curtate cycloid (radius B) as a fraction of the arc length of a regular cycloid (radius A)?
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1answer
42 views

Calculus Help Finding the Arc Length

So I'm having trouble with this problem: Let $$x=\frac{y^2+2y}{8}-\ln(y+1)$$ Find the arclength for $0\leq y\leq 2$. My work. I know the Arc length formula is $(1+ (x'^2))^{1/2}$ in this case ...
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1answer
36 views

Why are these two ways of measuring the length of the groove in a phonograph record different?

I once heard about the following problem on a math exam for students in about grade 8 or 9. A $33 \frac{1}{3}$ rpm record is 12 inches in diameter and a label diameter of 4 inches. If the groove ...
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2answers
66 views

How do I find the arc length between two points?

Let $c$ be the path $c(t)$ = $(t, 2sin(t), 3cos(t))$. Find the arc length of $c$ between the to points $(0, 0, 3)$ and $(\pi, 0, -3)$. I know the formula is the integral of the magnitude of the ...
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1answer
37 views

Is it possible to identify a circular arc given its length and two of its endpoints?

Suppose you have the length of the circular arc AB, in addition to the coordinates of A and B. Is this information sufficient to draw the arc (i.e. find its center point)?
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1answer
41 views

How do I prove the circumference of an ellipse

I'm trying to find the circumference of an ellipse with a horizontal radius of $h$ and a vertical radius of $k$. The equation for such an ellipse centered at the origin would by $(x/h)^2 + (y/k)^2 = 1$...
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2answers
37 views

Use calculus to find length of y=-mx+b, then show that the answer agrees with the answer when using pythagorean theorem

First question, sorry for poor formatting. This question is from my Calculus 2 class, and I am pretty sure I am supposed to be using arc length formula for the question. Exact words: Consider the ...
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1answer
34 views

Alternative Formulation of Arc Length

If $\gamma : [a,b] \to \Bbb{R}^n$ continuously differentiable, show that the arc length $arc(\gamma)$ equals $\sup \{ \sum_{i=1}^n ||\gamma (t_i)-\gamma (t_{i-1})|| \mid a = t_0 < t_1 < ... < ...
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1answer
47 views

Determine the length of the Parametric Curve given by the set of parametric equations.

I am seeking validation for my answer for the given problem below. Question: "Determine the length of the Parametric Curve given by the set of parametric equations." Parametric Equations: $x = 3 + ...
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2answers
33 views

Equation for arc with decaying radius

Hoping for some insight into the equation and mechanics of an arc with a decaying radius. Say at 0 degrees / 0 rad, the radius ...
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0answers
42 views

Integration by substitution to find the arc length of an ellipse in polar form.

I have that $l/r = 1+e.\cos(x)$, for $l = a(1-e^2)$ (constant). The question asks for the mean distance over angle of the planet from the sun, where the planet moves on an elliptical orbit with the ...
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3answers
80 views

Calculate the length of the closed curve $x^{2/3} + y^{2/3} = 4$

I realise that this function forms a closed curve, and the range of both $x$ and $y$ are: $-8 \leq x, y \leq 8$. I began by differentiating the function implicitly, arriving at a expression for $\...
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1answer
37 views

How do I complete this arc length problem?

Let $c(t) = (t^3, t^2, 2t)$ and $f(x,y,z)=(x^2-y^2, 2xy, z^2).$ (a) Find $(f\circ c)(t)$ (b) Find a parametrization for the tangent line to the curve $f\circ c$ at $t=1$. I know how to find the ...
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0answers
101 views

How to calculate arc distance on a sphere

I hope my question makes sense. I just don't know how to describe it using math lingo. Please bear with me. Let's say on a globe I'm traveling from point A to point B that is exactly opposite side of ...
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1answer
36 views

Proving arc length polar coordinate formula

I take the definition of arc length of a smooth curve between $x=a$ and $x=b$ to be: $$\int^a_b \sqrt{1+f'(x)^2}dx$$ Then how do I derive the formula for polar coordinates? And please don't use ...
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1answer
60 views

Length of a shadow cast by an object on a sphere

Say there is an object with height y standing on a spherical globe with radius r. A light ray casts a shadow from the object to the ground at angle θs. How can I find the length of the shadow d that ...
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114 views

Affine arc length

I was looking for an analog of arc length for plane curves in affine geometry, but I have only found the equi-affine arc length $d\sigma ={ || \gamma '(t)\wedge \gamma ''(t) ||}^{1 \over 3}dt$. On ...
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Inverse Arc Length of a Parabola

Is the inverse for the arc length of a parabola (say, $f(x)=\dfrac{x^2}{2}$) not discovered, or not possible to express given elementary functions and product log ($W(x)$)? If the latter is so, is ...
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0answers
1k views

How to determine linear terms from the nonlinear dataset?

Let us take the parametric curve r($t$) = [$t^2$;$t$], $t$ = [0,1]. Using this equation, I generate 1000 points. Now my goal is to determine the value of $t$ for each point on the curve without using ...
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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 + (\...
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Find the length of connected arcs

There are square tiles $20$ cm $\times$ $20$cm which are to be laid on a table top with dimensions of $80$cm $\times$ $80$cm. The design on the tile consists of two arcs of circles on the opposite ...
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1answer
73 views

Calculating arc length of a curve by pythagorean theorem

I was asked to explain the parametric curve arc length to a fellow student, only to find out that I don't completely understand it myself to be able to explain it. I've read multiple posts here about ...
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59 views

Inscribed triangle lengths from arc length

Consider a circular arc, radius $r$, with endpoints $A$ and $D$. Arc $AD$ subtends an angle $\varphi$ < $\pi$. Thus, the arc length of $AD$ is $s_{AD} = r\varphi$. Place two points, $B$ and $C$ on $...
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51 views

how to get the center of moved equilateral triangle according to endpoints displacement?

sorry if I did not use the proper jargon because I can't recall any specific words. $\mathbf Conditions:$ There is an equilateral $\Delta ABC$ in $\Bbb{R^3}$ with given side-length which lies on $...
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2answers
39 views

Arc Length of a Curve given point in (x,y,z) form

$r(t)=\sqrt{2}t$$i$+$\sqrt{2}t$$j$+($1$-$t$$^2$)$k$ Is the function I am considering, and I am finding the arc length. That is not what I am stuck on though, I am asked to find the arc length from $(...
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0answers
42 views

Finding the length of the arc.

Find the arc length of $$r\langle t\rangle = \langle e^{t^2},3e^{t^2}-2,\frac{3}{2}e^{t^2}\rangle\text{ for } 0\le t\le 1$$ My try: Arc length $= \int | r^1 (t) | \, dt$ $$ r^1(t) = \langle 2te^{...