Arc length Definition and 281 Threads

Several authors state the formula for finding the arc length of a curve defined by ##y = f(x)## from ##x=a## to ##x=b## as: $$\int ds = \int_a^b \sqrt{1+(\frac{dy}{dx})^2}dx$$ Isn't this notation technically wrong, since the RHS is a definite integral, and the LHS is an indefinite integral...

I'm doing a catapult project but I'm sort of confused. I need to find the spring constant in order to get the elastic potential energy. The force of pulling back the catapult lever to 36 degrees above the horizontal is 4.2 N. Right before the lever is at rest, 90 degrees, the force is 1.4 N. One...

A circle has a radius of 10cm. Find the length s of the arc intercepted by a central angle of 124° . Do not round any intermediate computations, and round your answer to the nearest tenth. How do I do this?

Homework Statement My class is working through chapter 2 of Newman's Analytic Number Theory text (on partitions). We have come to a part where he states that "elementary geometry gives the formula" (for the length of arc A) 4r\text{arcsin}\frac{\sqrt(2)(1-r)}{\sqrt(r)} We are attempting to...

Hi, So I'm working through a bunch of problems involving gradient vectors and derivatives to try to better understand it all, and one specific thing is giving me trouble. I have a general function that defines a change in Temperature with respect to position (x,y). So for example, dT/dt would...

Homework Statement Find the arc length parametrization of the curve r = (3t cost, 3tsint, 2sqrt(2)t^(3/2) ) . Homework Equations s(t)=integral of |r'(t)| dt The Attempt at a Solution I was able to get the integral of the magnitude of the velocity vector to simplify to: s(t) = integral of...

Now i haven't checked yet whether or not this is correct, but the formula for the length of an arc that subtends a central angle can also be expressed this way: AC/360 Where: A: Central Angle C: Circumference Is this correct? Thank you for your help.

Homework Statement Find the exact length of the curve: y= 1/4 x2-1/2 ln(x) where 1<=x<=2 Homework Equations Using the Length formula (Leibniz) given in my book, L=Int[a,b] sqrt(1+(dy/dx)2) I found derivative of f to be (x2-1)/2x does that look correct? The Attempt at a Solution I found f'...

Homework Statement A curve has the equation y2 = x3. Find the length of the arc joining (1, - 1) to (1, 1). Homework Equations The Attempt at a Solution I took the integral of the distance and tried to evaluate from -1 to 1. L = [intergral (-1 to 1) sqrt (1+(dy/dx x^3/23/2)2 dx] Evaluated I...

Homework Statement Find the length of the path traced out by a particle moving on a curve according to the given equation during the time interval specified in each case. The equation is r(t) = a(cos t + t sin t)i + a(sin t - t Cos t)j, 0</=t</=2pi, a>0Homework Equations Arc length =...

let C be the curve of intersection of the parabolic cylinder $x^2=2y$ and the surface $3z=xy$. find the exact length of C from the origin to the point (6,18,36). please help! this is the last question i have left from this assignment and i have no idea how to do it. i have grading to do and a...

Homework Statement Find the arc length of one of the leaves of the polar curve r= 6 cos 6θ. Homework Equations L = ∫sqrt(r^2 + (dr/dθ)^2) dθ (I use twice that since the length from 0 to π/12 is only half the petal) The Attempt at a Solution I seem to get an integral that can't be...

Of course, I need to find the first derivative and integrate its norm. α'(t) = (1, 0, (1/2)t^2 - (1/2)t^-2) ∫ [1 + (1/4)t^4 + (1/4)t^-4]^(1/2) dt, t = 1 to t = 3. Have I simply forgotten useful integrals? α'(t) = (e^t, -e^-t, root2) ∫ [e^2u + e^-2u + 2]^(1/2) du, u = 0 to u = t.

EDIT: Okay now that the admin has cleaned up my mess, please scroll down to see the correct image and the question on the 3rd post in this thread.

find the arc length function for the curve $y=2x^{3/2}$ with starting point $P_{0}(1,2)$. how do i do this? this is what I've done so far. $y'=3\sqrt{x}$ $1+(3\sqrt{x})^2=9x+1$ $\int_{a}^{b} \ \sqrt{9x+1},dx$ what's my a and what's my b?

how do i use the nspire to find arc length?

I learned in my calc 1 class that to calculate the arc length of a curve, we are to compute the integral of the function. For example, the integral of a function that describes the path of a thrown baseball would give the total distance traveled by the baseball (I hope I'm using the term arc...

1.) The problem is: Find the arc length of f(x)= x^3/3-1/(4x) from x=1 to 2 2.) Relevant formulas: ds = √(1+(dy/dx)) abs(L) = ∫ds 3.) My work so far: f'(x)= x^2+1/(4x^2) abs(L) = ∫(from 1 to 2) √(1+(x^2+1/(4x^2))^2 dx = ∫(from 1 to 2) √(1+(x^4+1/2+1/(16x^4)) dx = ∫(from 1...

So here's a little background for the question: I have an arc that covers 3/4s of a circle (so it's not quite a full circumference) such that the radius from the center of the arc varies with respect to the angle (dR/d(theta)) (and it can be either positive or negative, but not constant). I am...

Homework Statement Find the length of the curve x = 3 y^{4/3}-\frac{3}{32}y^{2/3}, \quad -64\le y\le 64Homework Equations Integral for arc length (L): L = \int_a^b \sqrt{1 + (\frac{dy}{dx})^{2}} dx The Attempt at a Solution Using symmetry of the interval and the above integral for arc length...

∫sqrt(x^4/4 + 1/(x^4) + 1/2) dx from x = 1 to 4 Could someone help me solve this? I can't seem to find a substitution that works, or find the square root of (x^4/4 + 1/(x^4). Any help would be very appreciated. Thanks in advance!

This may seem like a dumb question, but is finding the "arc length" of a curve and finding the "length" of a curve the same thing? Just worded differently?

Hello, I solved the arc length for a particular problem. However, what is the unit of arc length if the units of the velocity vs time graph are m/s vs s? I am really confused.

1. The problem statement, all variables and given/known If C is a smooth curve given by r(s)= x(s)i + y(s)j + z(s)k Where s is the arc length parameter. Then ||r'(s)|| = 1. My professor has stated that this is true for all cases the magnitude of r'(s) will always equal 1. But he wants me...

Find the length of the positive arc of the curve y=cosh^{-1}(x) (for which y≥0) between x=1 and x=\sqrt{5}. My attempt: x=cosh(y) → \frac{dx}{dy} = sinh(y) → (\frac{dx}{dy})^{2}=sinh^{2}(y), so ds=dy\sqrt{1+sinh^{2}(y)}, therefore the arc length is S=\int_{y=0}^{y=cosh^{-1}(\sqrt{5})} cosh(y)...

The solution to this question (whose answer is pi) is eluding me: The radius of a circle is 3 feet. Find the approximate length of an arc of this circle, if the length of the chord of the arc is 3 feet also.

how do i integrate the function sqrt(1 + 1/2(y^1/2 - y^(-1/2))^2) from 0 to 1??

Homework Statement The length L or the curve given by \frac{3y^{4}}{2}+\frac{1}{48y^{2}}-5 from y=1 to y=2 Homework Equations The Attempt at a Solution Setting up the formula is easy. First I found the derivative of f(y) which is: f'(y)=6y^{3}-\frac{1}{24y^{3}} Then I plugged...

Homework Statement I am currently reviewing the physics of 'standing waves on a string'. I know that for the nth harmonic, the length of the 'string' is \frac{n\lambda}{2}. Instead of just memorizing these, I have been trying to apply my knowledge of Calculus to figure out why these numbers...

Ello every one, i have interesting question. Any one who has james stewert 7th edition calc book I am on secotion 8.1 studying for an exam. number 13 of 8.1 says this y= ln(secx) find arc length from 0-pi/4 here is what i do first in my opinion. y`= 1/sec(sectan) y`= tanx...

Homework Statement Calculate the arc length of <2t,t^2,lnt> from 1=<t=<e Homework Equations Arc length=∫√{(x')^2 + (y')^2 + (z')^2} The Attempt at a Solution So I have gotten to this point: ∫√{4 + 4t^2 + \frac{1}{t^2}} Am I on the right track, and if so, how do I integrate that?

This is from my course notes http://img28.imageshack.us/img28/2630/ckyl.jpg In line 3, there's the integral \int_0^t ||y'(s)||ds which represents the length of the curve as a function of t (which I am thinking of as time). Here, I think s is a dummy variable for time. The equation in line...

Homework Statement Find the arc length parameterization of r(t) = <(e^t)sin(t),(e^t)cos(t),10e^t>The Attempt at a Solution so I guess i'll start by taking the derivative of r(t)... r'(t) = <e^t*cos(t) + e^t*sin(t), -e^t*sin(t) + e^t*cos(t), 10e^t> ehh... now do I do ds = |r'(t)|dt and...

Homework Statement Find the arc length of polar curve 9+9cosθ Homework Equations L = integral of sqrt(r^2 + (dr/dθ)^2 dθ dr/dθ = -9sinθ r = 9+9cosθ )The Attempt at a Solution 1. Simplifying the integral r^2 = (9+9cosθ^2) = 81 +162cosθ + 81cos^2(θ) (dr/dθ)^2 = 81sin^2(θ)...

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

Given t\in Ithe arc length of a regular parametrized curve \alpha : I \to \mathbb{R}^3 from the point t_0 is by definition s(t) = \int^t_{t_0}|\alpha'(t)|dt where |\alpha'(t)| = \sqrt{(x'(t))^2+(y'(t))^2+(z'(t))^2} is the length of the vector \alpha'(t). Since \alpha'(t) \ne 0 the arc length s...

Given t\in Ithe arc length of a regular parametrized curve \alpha : I \to \mathbb{R}^3 from the point t_0 is by definition s(t) = \int^t_{t_0}|\alpha'(t)|dt where |\alpha'(t)| = \sqrt{(x'(t))^2+(y'(t))^2+(z'(t))^2} is the length of the vector \alpha'(t). Since \alpha'(t) \ne 0 the arc length s...

Homework Statement Find the arc length of the graph, on the interval [1/2, 2], of y = \frac{x^3}{6} + \frac{1}{2x} Homework Equations s = \int^b_a \sqrt{1 + [f'(x)]^2}dx The Attempt at a Solution I began with s = \int_{1/2}^2 \sqrt{1 + (\frac{x^2}{2} - \frac{1}{2x^2})^2}dx...

hello every body .. According to the picture: Circle radius (Radius) and height (High) is known to us. Given that the height of the draw the tangent line , I looking for the equation for length of the arc (Arc Length) was calculated based on height changes. (sorry for my written...

Homework Statement Find the arc length of the curve (t) = (1; 3t2; t3) over the interval 0  t  1. Homework Equations L=sqrt(f'(t)^2+g'(t)^2+...+n'(t)^2) (integrated from a to b) int(udv)=uv-int(vdu) The Attempt at a Solution Seems like it should be fairly straightforward-- the...

Homework Statement Joe is traveling from point A across a circular lake to a cabin on the other side at point B. The straight line distance from A to B is 3 miles and is the diameter of the lake. He travels in a canoe on a straight line from A to C. She then takes the circular trail from C to...

The centerfield fence at a ballpark is 10 ft high and 400 ft from home plate. The ball is 3 ft above the ground when hit, and leaves with an angle theta degrees with the horizontal. The bat speed is 100 mph. Use the parametric equations x = (v0cos(theta))t y = h + (v0sin(theta))t - 16t^2 a...

This is an example in book by Howard Anton: Vector form of line is ##\vec r=\vec r_0+t\vec v## where ##\vec v## is parallel with the line. So both ##\vec r## and ##\vec r_0## are POSITION VECTORS. To change parameters, 1)Let u=t ##\Rightarrow\; \vec r=\vec r_0+u\vec v##. 2) ##\frac {d\vec...

Calculate the length of the curve We got the formula $$\int_a^b\sqrt{1+[f'(x)]^2}$$ and $$f'(x)=\frac{x}{36}-\frac{9}{x} <=> \frac{x^2-324}{36x}$$ so now we got $$\int_9^{9e}\sqrt{1+(\frac{x^2-324}{36x})^2}$$ we can rewrite that as $$\int_9^{9e}\sqrt{1+\frac{(x^2-324)^2}{1296x^2}}$$ then do...

Homework Statement Show that \gamma : [a, b] \rightarrow \Re^{2} is a parameterization of \Gamma if and only if the length of the curve from \gamma(a) to \gamma(s) is s - a; i.e., \int ^{s}_{a} \left| \gamma ' (t) \right| dt = s - a. Homework Equations The Attempt at a Solution Part 1...

Homework Statement Prove that any curve \Gamma can be parameterized by arc length. Homework Equations Hint: If η is any parameterization (of \Gamma I am guessing), let h(s) = \int^{s}_{a} \left| \eta ' (t) \right| dt and consider \gamma = \eta \circ h^{-1}. The Attempt at a Solution...

Homework Statement Explain why ∫(1+(1/x2)1/2dx over [1,e] = ∫(1+e2x)1/2dx over [0,1] The Attempt at a Solution The two original functions are ln(x) and ex and are both symmetrical about the line y = x. If I take either of the functions and translate it over the line y = x the two...

Homework Statement Homework Equations The arc length equation? The Attempt at a Solution I don't know where to begin.

Homework Statement Find the Arc Length from (0,3) clockwise to (2,sqrt(5)) along the circle defined by x2 + y2 = 9 Homework Equations Arc Length formula for integrals The Attempt at a Solution I have the correct answer at 3arcsin(2/3), but I tried to do this without calculus the first time...

Could someone please explain why PQ in the diagram below is rΔθ? Isn't rΔθ arc length? The best reason I can think of is that it's only an approximation for when the angle is very small, so PQ≈arclength=rΔθ. Not 100% sure though. http://imageshack.us/scaled/landing/199/feynmanangle.jpg...