Line integral Definition and 404 Threads

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 is used 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). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as



W
=

F

⋅

s



{\displaystyle W=\mathbf {F} \cdot \mathbf {s} }
, have natural continuous analogues in terms of line integrals, in this case




W
=

∫

L



F

(

s

)
⋅
d

s




{\displaystyle \textstyle W=\int _{L}\mathbf {F} (\mathbf {s} )\cdot d\mathbf {s} }
, which computes the work done on an object moving through an electric or gravitational field F along a path



L


{\displaystyle L}
.

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  1. S

    General meaning of line integral in vector fields

    So, as i understand, the geometrical meaning of this type of integral should still be the area under the curve, however, I really do not see how you can obtain each infinitesimal rectangle from the dot product. I have understood the typical work example, that is, the line integral as the sum...
  2. J

    Line integral, incorrect setup

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  3. B

    Line integral in a uniform force field

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  4. P

    MHB Stefan's question about a line integral.

    The line starting at $\displaystyle \begin{align*} \left( 0, 0, 0 \right) \end{align*}$ and ending at $\displaystyle \begin{align*} \left( \frac{1}{3}, \frac{\pi}{2}, 1 \right) \end{align*}$ can be expressed in a parametric (vector) form as $\displaystyle \begin{align*} \left( x, y, z \right) =...
  5. J

    Why is preferable the line integral than the area integral over C plan

    Why is preferable to use the line integral than the area integral over the complex plane?
  6. G

    How you can say if a line integral will be independant ot a given path

    Homework Statement Here is my problem : so far I've solved the line integral but I don't know what is the condition that must be met in order to be independant of the path given. I found the line integral to be: 27/28
  7. G

    Line integral of straight lines path (quick question)

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  8. A

    Volume integral turned in to surface + line integral?

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  9. J

    Why inverse laplace is line integral?

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  10. BiGyElLoWhAt

    Evaluate the Line Integral ∫_{C}xyds for x=t^2 and y=2t, 0≤t≤5

    ok, my turn to ask a question. Problem: evaluate ∫_{C}xyds for x=t^2 and y = 2t from 0\leq t \leq 5 not sure what I did wrong, but here it goes: solve for ds: ds =\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2} = \sqrt{4t^2+4}=2\sqrt{t^2+1} substitute: ∫_{0}^5 4t^3\sqrt{t^2+1}dt...
  11. G

    How Is the Parabolic Path y=x^2 Related to the Line Integral Calculation?

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  12. O

    Line integral: how can it be > 0?

    Using my understanding of calculus, I don't understand why line integrals in 3-d space can give a result > 0. You are following a line and integrating under that line. The line has some length. But according to my understanding of calculus, it does not have a width. What is this arbitrary...
  13. M

    Line integral and greens theorem

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  14. mesa

    Question about line integral of F dot dr

    Homework Statement Evaluate the line integral of F dot dr where f(x,y)=<3x^2,2x+y> and C is a straight line segment from (1,2) to (5,4) Homework Equations Unfortunately I was out with family obligations when we covered line integrals and surface integrals so am stuck with the textbook...
  15. PsychonautQQ

    Vector Line Integral Homework: ∫F ds

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  16. I

    How do I parametrize a line integral with vector functions?

    (disregard the [5+5+5] in the question)attempt: dr=(et(cost)+(sint)et)\hat{i} + (-et(sint)+(cost)et)\hat{j} ∫<3+2xy, x2-3y2>\cdot<et(cost)+(sint)et, -et(sint)+(cost)et>dt ..at which point i remembered i had to parametrize F in terms of t, but didn't know how to do
  17. P

    Curl & Line Integral of Vector Field: Calculations & Results

    Homework Statement Given a vector field F=-y/(x^2+y^2) i +x/(x^2 +y^2) Calculate the curl of it the line integral of it in a unit circle centered at O Homework Equations The Attempt at a Solution I calculated that the curl is 0 but the line integral is 2π. I don't think this...
  18. A

    Can someone just check if I did this line integral correctly?

    Homework Statement Homework Equations The Attempt at a Solution $$\int _{ 0 }^{ 2\pi }{ xdx } \\ =-\int _{ 0 }^{ 2\pi }{ sintcostdt } \\ =0$$ It feels wrong.
  19. E

    Calculate complex integral as line integral

    Homework Statement We need to calculate this complex integral as line integral: Homework Equations The Attempt at a Solution This is correct, I guess: But not sure about this part: Are dx, dy, x, y chages correct or there is other method to use?
  20. G

    Solving DS for Line Integral: 5x^2 + 3y^2 = 4

    Homework Statement Say I have a line integral which I have simplified to: \int\int x+y dS Over some surface S, let's say 5x^2 + 3y^2 = 4 or something. Having arrived at this step, how do I determine dS? The formulas and methods we've been taught doesn't really lead to this step all...
  21. PeteyCoco

    Line integral of a spherical vector field over cartesian path

    Homework Statement Compute the line integral of \vec{v} = (rcos^{2}\theta)\widehat{r} - (rcos\theta sin\theta)\widehat{\theta} + 3r\widehat{\phi} over the line from (0,1,0) to (0,1,2) (in Cartesian coordinates) The Attempt at a Solution Well, I expressed the path as a...
  22. J

    Help solving line integral question

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  23. T

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  24. FeDeX_LaTeX

    Line Integral Problem: Calculating ∫(x+y)dx+(y-x)dy Along Various Curves

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  25. M

    Line Integral Calculations: Understanding Direction and Parametrization

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  26. S

    Line integral over a Vector Field

    Homework Statement Given a vector field F(x,y,z) = (yz + 3x^{2})\hat{i} + xz\hat{j} + xy\hat{k} Calculate the line integral ∫_{A}^{B}F\bullet dl where A = (0,1,3) and B = (1,2,2) Homework Equations Right, first of all, what is dl ? I've gone over all my course notes and...
  27. J

    Line integral around a circle in polar coordinates

    I know that \oint_{C}\mathrm{d}\vec{l} = 0, for any closed curve C. But when i try to calculate the integral around the unit circle in polar coordinates, I get a result different from zero. Here is my approach : \oint_{C}\mathrm{d}\vec{l} = \int_{0}^{2\pi}\hat{\phi}\mathrm{d}\phi =...
  28. S

    Find Mass and C.O.M using a line integral

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  29. S

    (Line integral) Compute work through vector field

    Homework Statement "Consider the Vector field F(x,y)=<cos(sin(x)+y)cos(x)+e^x, cos(sin(x)+y)+y>. Compute the work done as you traverse the Archimedes spiral (r=θ) from (x,y)=(0,0) to (x,y)=(2∏,0). (Hint: check to see if the vector field is conservative) Homework Equations 1) F(x,y)=<P,Q>...
  30. D

    MHB Can Different Parameterizations Yield the Same Line Integral Value?

    1. Consider the curve c= (x(t),y(t),z(t)) in space as t varies over [0, T ]. We could also parameterize this curve by c= x(τ^2 ),y(τ^2 ),z(τ^2) τ ∈ [0, sqrt(T)]. Show that one obtains the same value for the line integral using either parameterization. The line integral is just the integral for...
  31. N

    (Calc 3) Finding mass of a wire using line integral

    Homework Statement Find the mass of a wire in the shape of the parabola y=x2 for 1 \leq x\leq2 and with density p(x,y)=x. Homework Equations The Attempt at a Solution I just want to make sure I am setting this integral up right. Here is what I did: I parameterized the equation to x=t, y=t2...
  32. G

    Reducing surface integral to line integral?

    Homework Statement hi I am trying to adjust this general integral to my problem, my problem consists of a semi-infinite rod, i.e. x in [0,∞) the primed variables are the integration variables Homework Equations http://img339.imageshack.us/img339/5038/42247711.jpg The Attempt at a...
  33. V

    Complex line integral over x + y = 1

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  34. Y

    Verifying Line Integrals Using Vector Value Functions

    I want to verify I am doing this correctly first: Evaluate##\int_c (x^2ydx+xdy)## where the line is from (1,2) to (0.0) My method is different from the book, I am using vector value function method where ##<x(t),y(t)>-(x_0,y_0>=t\frac {d\vec r}{dt}## and ##\vec r=\hat x x(t)+\hat y y(t)##...
  35. U

    Is There a Mistake in Determining Conservative Fields?

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  36. D

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  37. idir93

    Line Integral Along a Path: How to Compute and Use Vector Fields

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  38. Y

    What Are the Differences in Line Integrals Using Different Coordinate Systems?

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  39. C

    Vector line integral notation.

    Hey, I'm studying for a physics degree and have a general curiosity about vector calculus. Having learned about surface and line integrals for scalar functions in multivariable calculus I've been having some issues translating them into vector calculus. Though conceptually I haven't had much...
  40. F

    Circular Helix Line Integral: Solving with r and dr/dλ

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  41. S

    Use Stokes's on Line Integral to Show Path Independence

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  42. K

    Voltage difference, line integral

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  43. C

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  44. STEMucator

    Computing another line integral

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  45. STEMucator

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  46. E

    Line Integral Homework: Solving Problems with W = F*dr and Pictures

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  47. U

    Is this Line Integral Independent of Path for a Conservative Field?

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  48. D

    Exploring Vector Field Line Integrals: A Sample Final Exam Problem

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  49. M

    Line integral of a vector field over a square curve

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  50. R

    Line Integral of Scalar Field Along a Curve

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