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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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Find the line integral. Have I set this problem up correctly?

F, the vector field, $ = [xy, x^{2}y^{2}]$ C is a quarter-circle from (2,0) to (0,2) with center (0,0). So r(t) = the parametric equation of C $=[2cost(t), 2sin(t)$ $$r'(t) = [-2sin{t}, 2cos{t}]$$ $$F(r(t)) = [4sin(t)cos(t), 4cos^{2}{t} *…
Jwan622
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Smooth curves and Line Integrals

Let $f:R^3->R$ be a continuous, and let $C_{1},C_{2}$ be two smooth curves joining two points $p,q$. Then prove that line integrals of $f$ with respect to $C_{1}$ and $C_{2}$ are equal. My progress: To be honest, I am quite unsure whether this…
BlizzardWalker
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what is actually calculated on the line integral?

I know from vector calculus, that line integral is the area of the curtain under the curve. Then, i'm realize we can solve the line integral with respect to $x$ or $y$ . Integration with respect to $y$ is just like ordinary definite integral. But…
user516076
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(Edited) Confusing on proving reverse direction of Line Integral

Now i'm taking complex varibles course and learning about complex integration. But this is still beginning and i need to improve my knowledge about line integral. (I forgot a little about line integral and its parametrization on vector analysis, so…
user516076
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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.
Wouter Lommerse
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How to find a special integrating factor which is not in form of $x^m y^n$ but is both function of $x$ and $y$?

I was watching this youtube video. Now I would wonder what if special integrating factor had been a function of $x$ and $y$ other than some exponential function of $x$ and $y$. Then how should we get that? suppose you have this differential…
user187604
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Line integral computing methods

Evaluate $$\int_L \frac{ds}{\sqrt{x^2+y^2+4}}$$ where $L$ is a straight line segment between points $A(0;0)$ and $B(1;2)$ Can anyone give me a hint how to solve this integral?
Martins2018
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Finding non-closed paths so that a line integral equals certain values?

So I'm required to find a non-closed path $C$ so that the line integral $$\displaystyle \int_C \mathbf{F} \cdot \mathrm d \mathbf{r}$$ equals $0$, and another non-closed path $C$ so that the line integral equals $2$. It's given that…
tsp216
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Finding the range of the parameter after parameterizing a line segment or a curve

I have these two planes: $x-y-z=0$ and $x+y+2z=o$ and I want to parameterize the line of intersection which is $x=3y$ to calculate the line integral from the origin to the point $(3,1,-2)$. $$\text{Parameterization: }\ x=3t,\, y=t,\, z=-2t$$ Here…
F.O
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Evaluate Integral over conservative vector field

Evaluate the line integral $$ \int\limits_C F \cdot dr $$ where $\DeclareMathOperator{grad}{grad}F= \grad f$, $f(x,y,z)=\sin(x)\cos(y)\,z$ and $C$ is the circle $x^2 +y^2=1$ and $z=3$. I understand that the value of the integral is zero because…
Eddy Zavala
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Evaluate $\int_c\tan ydx+x\sec ^2ydy$ where $C$ is any path from $(1,0)$ to $(2,\frac{\pi }{4})$

I got an answer of $\frac{4}{3}$, but the textbook answer says $2$. How would one solve it, and what did I do wrong? Let $\vec r=(1+t^2)\hat i+\arctan(t)\hat j$ where $0≤t≤1$ $dx=2tdt, dy=\frac{1}{1+t^2}dt, \sec^2y=1-\tan^2y$ $\int_c\tan ydx+x\sec…
David
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Compute the scalar line integral.

Compute the scalar line integral for () = on , where is oriented counterclockwise and comprises : the ellipse $\ \frac{^}{ }+ \frac{^}{}= $ in the first quadrant, : the line segment from $\ (, )$ to $\ (, )$
Mo Bhuiyan
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Evaluate the area integral $I=\int\mathbf{G}(\mathbf{r})\cdotp d\mathbf{A}$

A vector field $\mathbf{G}(\mathbf{r}) = yz\mathbf{i}+\mathbf{j}+x^2\mathbf{k}$ fill all the space. Evaluate the area integral $I=\int\mathbf{G}(\mathbf{r})\cdotp d\mathbf{A}$ over the rectangle in the $(x,y)$ plane with corners $ (0,0,0), (1,2,0),…
Stratos1
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simple line integral - scalar and vector

Evaluate the following line integrals, showing your working. The path of integration in each case is anticlockwise around the four sides of the square OABC in the x−y plane whose edges are aligned with the coordinate axes. The length of each side of…
Toby Peterken
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What is the line integral for $F=(y^2, x^2)$

When C is the curve along the sides on the triangle with corners in $(0,0)$, $(1,0)$ and $(0,1)$ with counter-clockwise (positive) direction. Then $\int_0^.Fdr$ is? Do I have to make a parameterization of the triangle? How do i go about…
J.Dawg
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