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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.

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; contour integral as well, although that is typically reserved for line integrals in the complex plane.

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).

Read more on wikipedia's entry Line integral.

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Line Integral Notation: What is r in f(r)?

I've had a tough time trying to interpret line integrals such as this one, which may be found on Wikipedia's Line Integral article: $$\int_C f(\mathbf r) \,ds$$ More specifically, I'm trying to understand what $\mathbf r$ is in this integral. I…
Alejandro Miller
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Can a function be line integrated over a discontinuous curve?

Given that $g(x,y)=x^3y+xy^3$, suppose we wish to compute $$\int_{C}{\nabla g\cdot\textrm{d}\mathbf{r}},$$ where $C$ is the contour line $g=5$. The solution given is that since the gradient is always orthogonal to contour lines, the line integral…
Andrew Paul
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Line integral along ellipse, but with only one variable in integrand

I have to calculate this integral: $$\int_Lyds$$ where L is a part of ellipse $$\begin{cases} x=2\cos(t) \\ y=3\sin(t) & \ \end{cases}$$ in first quadrant. The problem is the integrand contains only $y$ instead of $xy$. That's why "$u$…
Martins2018
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Find $\int_{L}\overrightarrow{F} \cdot d\overrightarrow{r}$

Question $\def\vec{\overrightarrow}$Let$$\vec{F} = \left(3+2xy\right)\hat{i}+\left(x^{2}-3y^{2}\right)\hat{j}$$ and let $L$ be the curve$$\vec{r}=e^{t}\sin t \hat{i}+e^{t}\cos t \hat{j}.$$ Then find $\int_{L}\vec{F} \cdot d\vec{r}$. MY…
Mohan Sharma
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Evaluate the integral $\int_c(3x-y)ds$

Evaluate the integral $\int_c(3x-y)ds$,Where $c$ is the line segment from $(1,2)$ to $(3,3)$ followed by the portion of circle $x^2+y^2=18$ from $(3,3)$ to $(3,-3)$ in clockwise sense. My try:For line segment $x(t)=1+2t$,$y(t)=2+t$ ant $0\leq t\leq…
MatheMagic
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Line Integrals given points

I am stuck on the following problem: Evaluate $\int_c xdx + ydy +zdz$ where $C$ is the line segment from $(4,1,1)$ to $(7,-2,4).$ I found the line equations (I believe that's what they're called) for $x, y,$ and $z$, getting $$x(t)=3t+4$$…
mathjohnn
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Evaluate the line integral over the vector field given this semi-circle

So here is the problem on the practice test verbatim Evaluate $ \int_{C}^{ } \textbf{F} \cdot d\textbf{r} \text{ where } C = \{ (x,y) \in \mathbb{R} \mid (x-1)^2 + y^2 = 1 \text{ and } y \ge 0 \} $ oriented counter clockwise and $\textbf{F}(x,y) = …
financial_physician
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Evaluation of this line integral $\int z^2\cdot e^\left(\frac{1}{z}\right)\cdot\sin\left(\frac{1}{z}\right)dz$

where the contour is $$|z|=1$$ if I use residual theorem $$res f(z) = \lim\limits_{z\to 0}  z^3\cdot e^\left(\frac{1}{z}\right)\cdot \sin\left(\frac{1}{z}\right)$$ this limit is undefined .However , if i expand the function in its laurent series i…
Heisenberg
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line integral Parametrization Problem

I am stuck with this question from my assignment based on line integral- What I don't understand here is that given c(t), I can directly plug in values of x, y and z (respectively cost, sint and t/2π) and solve the integral for 0 < t < 2π but than…
Vishal K Nair
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Integrating curve over a line segment

I'm having some trouble understanding this question: what is $\int_C xdy - ydx$, where C is the curve composed of a straight line segment C from $(−2, 0)$ to $(0, 0)$, a straight line segment from $(0, 0)$ to $(0, −2)$, and the part of the…
thbcm
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Line integrals and invariance of parameterization

The line integral of a scalar function $f(x,y)$ along a curve $\vec{r}(t)$ for $a \leq t \leq b$ is defined to be $$ \int\limits_{\vec{r}(t)} f(\vec{r}(t)) \, ds = \int_a^b f(\vec{r}(t)) \, ||\vec{r}\,'(t)|| \, dt $$ and my Calculus text says this…
Jason D.
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Why the result of this line integral depends on the parametrization? (it is not the orientation)

I'm tring to solve the following line integral: $$\int_C (2yx^2-4x) ds$$ where $C$ is the lower half of the circle centered at the origin of radius 3 with clockwise rotation. However, if I use the parametrization $r_1(t)=(3\cos(t),-3\sin(t))$,…
AlephZero
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Calculate the line integral using parameterization

I'm struggling with a question and I can't solve it, I can't find good parameterization so I will be able to calculate the line Integral directly. The question : Given the line integral $C$ : $y=x^3$ from $(0,0) \rightarrow (1,1)$. Calculate the…
Roach87
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Surface area using line integrals

Using line integrals and formula to calculate the surface area($\int zds$ on some curve $c$) I need to find area of cylinder between the plane $z=0$ and surface $z=R+\frac{x^2}{R^2}$. After parametric equation $x=R\cos t, y=R\sin t$ i get that…
Milica Koprivica
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Calculate line integral.

I have an example problem of a line integral below: It makes sense but the above problem is somewhat easy because the parametric equation is given to us. I have this next problem: Calculate $\int_{C} F(r) \cdot dr$ for the given data. $$F =…
Jwan622
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