Questions tagged [line-integrals]

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used.

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Please explain this identity

A reference from “introduction to calculus and analysis II/1” p. 89: $u=f(x,y,z)$, we are able now to define the integral $\int L$ of the linear differential form $$L=A(x,y,z)dx+B(x,y,z)dy+C(x,y,z)dz$...
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Help for verifying approach for executing a line integral of kind 2

I have a line integral of kind 2. I want to use Green's theorem to solve it. I am not sure if I am setting it right so I want to ask for help to verify if I set up the integral correct. So the ...
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Help getting a representation of a curve K for a Line integral

So I have to calculate line integral of a vector field: $\overrightarrow{F}(x,y,z)=(x+y,y+z,x+y)$ And I want to calculate the Line integral: $$\int_K \overrightarrow{F}(x,y,z)d\overrightarrow{r}$$ ...
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Verifying approach for solving Line integral of kind 1 ( Homework )

I am trying to solve a Line integral of 1nd kind given as follows : $$\int_C(x^2+y^2-2z)dl$$ on the curved line given as follows:$$C:x=4\cos(2t),y=4 \sin(2t),z=6t,t\in[0,3\pi].$$ So i am just using ...
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Help for evaluating a line integral with Green's theorem

I have the following line integral of kind 2 $$\iint (2x)dx+3(yx)dy$$ and the region $$C:4\cos(2t) \ , \ y=3\sin(2t)$$ I sketch the region and its an elipse: $$\frac{x^2}{4}+\frac{y^2}{3}=1$$i am ...
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The right way of using Green's theorem when x and y are given parametrized

I am solving a line integral that is from $2nd \ kind$ and it can be solved by direct integration or by green's theorem so i already solved it using the direct integration but i have tried different ...
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Is it legal to integrate $\sin(4x)$ and $\cos(2x)$ with $u$ substitution

I have a problem with solving one integral due to lack of experience in the integration logic. So i have line integral of kind 2 as follows: $$\int_0^{2 \pi}(2x) \ dx + (3yx) \ dy,$$ given by $C: x ...
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Help for understaning the formula for integration a Line integral of first kind

I need help to understand this formula: $$\int_Гf(x,y,z)dl= \int_{t1}^{t2}f(\varphi(t), \psi(t), \mathcal X(t)) \sqrt{(\varphi^\prime)^2t \ + \ (\psi^\prime)^2t \ + \ \mathcal (X^\prime)^2}dt$$ First ...
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Integral for a conservative vector field

I'd like some guidance on how to solve this problem. Given vector field: $$ \vec{F} = <y \cos(xy), x \cos(xy), \frac{1}{1+z^2} >$$ and I need to compute the integral $ \int_{C} \vec{F}.d\...
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Finding path for maximum value of line integral

I need help in the following question: I have a field: $$F=\left(y^3-3y+xy^2,3x-x^3+x^2y\right)$$ bounded in region $D$ defined by $x^2+y^2\leq 2.$ I need to find a path $C$ that goes from $(1,1)$ to ...
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Multivariable Chain Rule and Line Integrals Over Vector Fields

The process of solving an exact 1st order ODE is the inverse of applying the multivariable chain rule to a function, analogous to inverting the product rule to solve a linear 1st order ODE. For ...
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line integral of a function

I have to calculate the integral $$\int_{K}xy \cdot dx $$ along the curve K with equation $y=x^2$ at which $x$ varies from $1$ to $2$. I don't know how to begin answering this question.
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Physical unit of the line segment dS in a line ntegral

My question is about line integrals. Say we have a function $F(a,b)$ with units N/m/s. The units of $a$ is s and units of $b$ is m. To get the total force we could double integrate: $F_0=\int_{0}^{\...
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Explanation of coefficient when evaluating contour around a branch for fractional version of Cauchy's Integral Formula

I am working on fractional derivatives which are defined by taking the Cauchy Integral formula and letting the order of the derivative be non-integer. Specifically, \begin{equation} f^{(\alpha)}(z)=\...
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mean of a field along the line

What is the proof of a mean field along any line? Or how can we define it? i.e \begin{equation} a=\int_0^1 b \, dx \end{equation} where $a$ is the mean of $b$ and $b(0)=b(1)=0$.
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Question about line integrals with various options

I am new to calculus and am looking for some feedback regarding the following question. Many thanks in advance! The following line integral is given: $\int_C(y + y cos(xy))dx + (x + x cos(xy))dy$. ...
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Calculating the mass of the ellipse curve

I want to calculate the mass of the ellipse curve $\vec\gamma(t)=(acos(t),bsin(t))$ for $a>0,b>0$ and for $t \in[0,2\pi]$, where line density $\lambda(x_1,x_2)=Cx_1x_2$ for $C>0$. After ...
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Triangle parameterisation

I get how to answer the qs below, the problem is actually finding path $2$ $ \left( 2, 0, 0 \right) $ to $ \left( 0, 1, 0 \right) $ I get $(2-t)i+tj$ yet the answer for path 2 is... $$ (2-t)i+(t/2)...
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A given perimeter length that is circular encloses the maximum area - which are the (analytic) proofs? [duplicate]

I'm guessing Newton, because of his integrals. But what proofs have been established, and which is the most mathematically intuitive one? I was looking for the tag "circumference", supplied the newer ...
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How to compute the line integral of $\int_{C^+}(x^2-y)dx+(y^2+x)dy$?

I need help with this problem: Compute the line integral$\int_{C^+}(x^2-y)dx+(y^2+x)dy$ where $C^+$ is the parabollic arc $y=x^2+1$, $0\leq x\leq 1$ oriented from $(0,1)$ to $(1,2)$. First I ...
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How to prove the integral formulae of the inverse path $\alpha^-$ and the product path $\alpha\beta$?

I need help with this problem: Let $f:S\subset\mathbb{R}^n\rightarrow\mathbb{R}$ be continuous on $S$, and let $\alpha:[a,b]\subset\mathbb{R}\rightarrow\mathbb{R}^n$ and $\beta:[c,d]\subset\mathbb{...
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How to evaluate the line integral (checking Stokes Theorem)

Consider the vector field: $$\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k$$ A closed curve $C$ lies in the plane $x + y + z = 3$, oriented counterclockwise. The parametric ...
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Work done by three-dimensional inverse square field

I'm confused on how to set up the problem. Any help would be great, thank you in advance!
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Define the path integral when scalar $f$ and curve $\mathbf{c}$ is in *curvilinear* coordinates

I'm doing a multivariable calculus course at the moment. I've seen path integrals in the cartesian coordinate system as the following definition: Definition. The path integral of $f(x,y,z)$ along the ...
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44 views

Using line integrals to find total surface area

I am given the equation for the height of a fence, which is h(x,y,z) = 4 + ((sinx)/2) + (y/3) +z. The line integral of this function will give the lateral surface area of one side of the fence. ...
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Integration problem with full derivative

I am trying to find out what type of integration is used to solve this problem and what are the rules behind it. I am sorry, if this question is quite basic, but I do not study Mathematics. The ...
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What kind of integration is this and what is the rule behind it?

I am trying to find out what type of integration is used to solve this problem and what are the rules behind it. I am sorry, if this question is quite basic, but I do not study Mathematics. The ...
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LHS and RHS of Stokes' theorem not equal

Question: I am trying to test my understanding of Stokes' theorem by calculating the left and right hand side of the theorems equality by way of example and seeing whether they equal each other. They ...
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Why calculate the line integral for a vector field? *Without using work/physics*

For a while now I've been trying to find motivation and a good intuition behind the line integral for a vector field. This is the first time I'm learning this topic and I'm not interested in too much ...
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line element as a 1 form

I'm studying differential forms and I know how to manipulate all the equations. On trying to find a pictorial understanding, I am a bit stuck on the following. A one-form is suppose to assign a ...
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How to show that two parametrizations are representing the same line in plane?

(Two parameterizations): $$\vec r_1(t) = (t^3, t + 1), t ∈ [0, 1] $$ $$\vec r_2(t) = (t^6, t^2 + 1), t ∈ [0, 1]$$ How can I show that these two parameterizations represent the same line in plane? ...
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How to convert an integration of a curl to the surface integral?

$\displaystyle\int \left[ \nabla \times \dfrac {M(r')}{r} \right] d\tau =\oint \dfrac{1}{r} \left[ M(r')×da' \right]$ I came across this integral from the David Griffiths introduction to ...
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Evaluating $\oint_C \vec{F} \cdot d\vec{r}$

After trying a couple of times, but failing to find a way to solve these problems, I decided I should perhaps ask the people on this forum for help. Problem 1 Let $C$ be the curve $(x-1)^2+y^2=16$, ...
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path independence of a scalar field line integral

I had seen mathematical proofs of why it's true. But I couldn't wrap around my head with the intuition behind it. For a single variable function, $\begin{equation} \int_{a}^{b}f(x) \,dx = -\int_{b}^{...
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Finding the maximum value of $\oint_C \vec{F} \cdot d\vec{r}$ of a vector field

I'm stuck with trying to find the maximum value for $\oint_C \vec{F} \cdot d\vec{r}$ of the vector field $\vec{F}=\langle5z, x, y \rangle$ where $C$ is a simple closed curve in the plane $2x+3y+z=7$. ...
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Integrating Factors Via Minimization

I am trying to algorithmize an idea to find integrating factors for inexact differential equations via optimization. Currently, I'm sticking to simple cases that can already be solved by other means. ...
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For Green's theorem, why is the region of integration of the line integral a weird partial derivative character?

Why the weird $\partial{Q}$ notation for the integral region for Green's Theorem? $$\int_{\partial{Q}} W \cdot ds = \iint_Q \frac{\partial{g}}{\partial{x}} - \frac{\partial{f}}{\partial{y}} dx\ dy$$ ...
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Evaluate the line integral of a vector field around a square

I am asking this question because I believe the answer sheet I was given has an incorrect solution. The task is to evaluate (by hand!) the line integral of the vector field $\mathbf{F}(x,y) = x^2y^2 \...
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Shortcuts for $\int_{\kappa}F dx$

Let $F(x,y,z):=\begin{pmatrix} x^{2}+5y+3yz \\ 5x +3xz -2 \\ 3xy -4z \end{pmatrix}$ and $\kappa: [0, 2\pi] \to \mathbb R^{3}, t\mapsto\begin{pmatrix}\sin t\\ \cos t \\ t \end{pmatrix}$ I was asked ...
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Finding the value of $\int{F \cdot dr}$

Find the value of $\int{F \cdot \mathrm{d}r}$, where $$F(x,y) = \langle 5e^y+ye^x,e^x+5xe^y \rangle$$ and $$C: r(t) = \left\langle\sin\left(\frac{\pi t}{2}\right),\ln(t)\right\rangle; 1\le t\le2$$ So ...
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Line integral for surface area

Use a line integral to find the area of the surface that extends upward from the semicircle $y=\sqrt{4-x^2}$ in the $xy$-plane to the surface $z=3x^4y$. I know how to compute line integrals but I'm ...
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Help in evaluating this line integral

Let $\gamma (t)=1+e^{it}, 0\le t\le 2\pi$. I have to find the line integral of $\frac{1}{(z^2-1)}$ with respect to $\gamma$. My attempt : $\gamma$ is the unit circle in the complex plane centered at ...
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Parametrization of a spherical triangle?

I want to calculate an integral in the curve defined by a spherical triangle with vertices $\left(a,0,0\right),\left(0,a,0\right),\left(0,0,a\right)$ over the sphere of center $\left(0,0,0\right)$, ...
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Question about the continuity of the length of a continous parametrized curve

Any hint or demo to prove that the length of a continous parametrized curve defines a continous function in a normed space?
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Line integral path independence proof check

Find the work done by the force $F(x, y, z) = (x^4y^5, x^3)$ along the curve C given by the part of the graph of $y$ = $(x^3)$ from $(0, 0)$ to $(-1, -1)$. I first checked for independence, which ...
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Closed form solution to $\int_0^{\pi/4}\frac{e^{ic\sqrt{1+ br^2(\theta)}}}{\sqrt{1+ br^2(\theta)}}\,d\theta$

Is there closed form solution to this integral $$\int_0^{\pi/4}\frac{e^{ic\sqrt{1+ br^2(\theta)}}}{\sqrt{1+ br^2(\theta)}}\,d\theta$$ $r(\theta)=\frac{a}{\cos(\theta)}$ is radius vector from the ...
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Line integral, independence of path when to use it

Let $F(x,y) = (3x^2,4y^3)$. Determine the value of $\int_c F(x,y)\cdot \mathrm dr$, where $c$ is the path from $(0,1)$ to $(\pi,-1)$ along graph of $y=\cos x$. Is it good to always check for path ...
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Calculating the area of a surface given by a set $S$

$S=\{(x,y,z):x^2+y^2+z^2=4, (x-1)^2+y^2 \leq 1 \}$. $x^2+y^2+z^2=4 \iff \frac{x^2}{4}+\frac{y^2}{4} +\frac{z^2}{4}=1$ I'm not exactly sure what to parametrize the set $S$ by I thought of using ...
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define work $W=\int_C \mathbf{F}\cdot d\mathbf{r}$

We define work as follows : $$W=\int_C \mathbf{F}\cdot d\mathbf{r}$$ and we know that $\mathbf{F}(\mathbf{r}(t),\mathbf{v}(t),t)$ now can we write ? : $$W=\int_C \mathbf{F}\cdot d\mathbf{r}\overset{?}{...
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Line Integral Over a Vector Field of a set of points where a sphere intersects $2z$ $+$ $x$ $=$ $0$

One of my practice problems asks me to compute $\int_C zdx+xdy$ where $C$ is the set of points satisfying $$x^2+y^2+z^2=4 \quad\text{and}\quad 2z+x=0$$ where $C$ is oriented counterclockwise. If I ...