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. A

    Use Green's theorem to evaluate the line integral

    Homework Statement Use Green's theorem to evaluate the line integral: ∫y3 dx + (x3 + 3xy2) dy where C is the path along the graph of y=x3 from (0,0) to (1,1) and from (1,1) to (0,0) along the graph of y=x. 2. The attempt at a solution I've completed two integrals for both paths (y=x3 &...
  2. M

    Line Integral: Computing for $\int _1 ^2 V(x)dx$

    Suppose I have a vector V and I want to compute for the line integral from point (1,1,0) to point (2,2,0) and I take the path of the least distance (one that traces the identity function). The line integral is of the form: \int _a ^b \vec{V} \cdot d\vec{l} Where: x=y, \ d\vec{l}...
  3. A

    Is the Line Integral Independent of the Path?

    Homework Statement Evalutate \int_{C} (y2-3x2)dx + (2xy+2)dy, where C is a smooth curve from (0,1) to (1,3). 2. The attempt at a solution I've checked through my notes and text but can't find an example. I'd appreciate it if someone could help me get this started.
  4. M

    Line Integral of a complex function

    I'm trying to solve this integral as x-> Infinity \int \frac{dz}{8i + z^2} ...on the y=x line, but I have no idea what I'm doing. The book I'm using is less than helpful in this regard. I'm not supposed to use any complex analysis tools (Cauchy, etc), but just solve it as a line...
  5. A

    Green's Theorem to evaluate the line integral

    Homework Statement Use Green's Theorem to evaluate the line integral of the vector field F along the given positively oriented curve C. F(x,y) = <sin(x^3) +x^2(y), 3xy-(x)(y^2)+e^(y^2)> and C is the boundary of the region enclosed by the semicircle y = √(4-x^2) and the x-axis. Homework...
  6. A

    Evaluating Line Integral: ∫(x+2y)dx+(x^2)dy

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  7. U

    Using Green's Theorem to evaluate the line integral.

    Homework Statement Green's Theorem to evaluate the line following line integral, oriented clockwise. ∫xydx+(x^2+x)dy, where C is the path though points (-1,0);(1,0);(0,1) Homework Equations Geen's theorem: ∫F°DS=∫∫ \frac{F_2}{δx}-\frac{F_1}{δy} The Attempt at a Solution What...
  8. C

    Line integral across a vector field

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  9. C

    Line integral across a vector field

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  10. K

    Help understanding/evaluating line integral over a curve

    Evaluate the line integral, where C is the given curve. \int_{c} xy\:ds, when C: x=t^{2}, \ y=2t\ , \ 0\leq t\leq4 To solve this I should use the formula \int^{b}_{a} f(x(t),y(t))\sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2}}dt This gives me \int^{4}_{0}...
  11. N

    What is the Geometrical Interpretation of line integral?

    So I'm able to calculate them no problem. But the problem is, I don't really understand what they mean. For example, W=∫Fdr I understand that a vector field is something that defines a vector at every point. Then if we pick a curve in the field, we are integrating along that curve. So does...
  12. S

    Given value of a line integral, find line integral along different curves

    Given value of a line integral, find line integral along "different" curves Homework Statement I think I've got this figured out, so I'm just checking my answers: Suppose that \int_\gamma \vec{F}(\vec{r}) \cdot d\vec{r} = 17 , where \gamma is the oriented curve \vec{r}(t) = \cos{t} \vec{i}...
  13. S

    Solving Squirrel Work Problem with Line Integral Setup

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  14. B

    Who is the creator of line integral?

    Hi I'd like to know who is the mathematician or physicist that create line integral, I've always studied how calculate it, but Who can that be, I wonder? Thanks Alfonso.
  15. R

    Is This New Theorem Related to Stokes' Theorem?

    I met a proof problem that is as follows. ##\bf a = ∫_S d \bf a##, where S is the surface and ##\bf a ##is the vector area of it. Please proof that ##\bf a = \frac{1}2\oint \! \bf r \times d\bf l##, where integration is around the boundary line. Any help would be very appreciated!
  16. estro

    Calculating line integral using Stokes' and Gauss' theorems

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  17. F

    Is this line integral computation correct? (Green's Thm application)

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  18. N

    Analyzing Line Integral over Non-Exact Region |x|+|y|=4

    Analyzing an integral over a non-exact region for gamma defined by |x|+|y|=4 The following was similar to a problem on a calculus final that I got wrong. It is an extension of a problem in R.C. Buck "Advanced Calculus" on page 501. Similar to knowing the trick to integrating e^{|x|} (which...
  19. J

    Solving Scalar Line Integral with Vector {\bf{u}}

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  20. M

    Parametrizing and Evaluating a Line Integral on a Given Curve

    Homework Statement Evaluate ∫(x^3 + y^3)ds where C : r(t)=<e^t , e^(-t)>, 0 <= t <= ln2 c Homework Equations The Attempt at a Solution I tried to parametrize the integral and change ds to sqrt(e^(2t) + e^(-2t)) dt. I then change (x^3 + y^3) to...
  21. A

    How Do You Calculate This Complex Line Integral?

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  22. A

    MHB How to Compute the Line Integral Over a Piecewise Curve?

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  23. marellasunny

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  24. G

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  25. P

    Calculating Line Integral on Curve C (IR3)

    Consider that the curve C (IR3) is the intersection between x2+2z2=2 and y=1. Calculate the line integral: Note: The curve C is traversed one time in the counter-clockwise direction (seen from the origin of IR3). My attempt: http://i.imgur.com/5EsJO.png Can someone check? Thanks! :)
  26. P

    Work - Line Integral Homework: Calculating Work along C1 & C2 Curve

    Homework Statement I have exam tomorrow and there's a problem I don't know how to do. Consider the curve C1 (x=-y^2+3y) and C2 (x=0), both defined for y\in[0,3]. Calculate the work done by F(x,y)=(x,y^2) along the curve C=C1UC2 (retrograde direction). Homework Equations The...
  27. R

    Line integral - confusion on squares and square root terms

    line integral -- confusion on squares and square root terms Homework Statement Do you see where they have sqrt(16 sin^2t etc = 5? How do they get that, the answer should be 7, the square root of 16 is 4, sin^2 + cos^2 is 1 and the square root of 9 is 3, 3 + 4 = 7. It's like they're...
  28. Y

    Line integral in n dimension(Stokes' theorem)

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  29. DryRun

    Path Independence in Line Integrals: Simplifying Evaluation | Problem Attached

    Homework Statement I have attached the problem to the post.Homework Equations Properties of line integral. Path independence. The Attempt at a Solution I have shown that the path is independent, as: \partial P/\partial y = \partial Q/\partial x The problem is with the parametrization. I found...
  30. D

    Green's Theorem & Line Integral confusion

    Homework Statement a) Evaluate the work done by the force field F(x, y) = (3y^(2) + x)i + 4x^(3)j over the curve r(t) = e^(t)i + e^(3t)j, tε[0, ln(2)]. b) Using Green’s theorem, find the area enclosed by the curve r(t) and the segment that joins the points (1, 1) and (2, 8). c) Find the...
  31. P

    Are Vectors Coordinate Invariant in Special Relativity?

    In the line integral of work done by conservative forces, should we care about the direction of force and differential displacement if we choose different coordinate system? The formula of potential energy comes out to be different if we choose different coordinate system.
  32. N

    Closed Line Integral Homework - Computing a Hypotenuse

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  33. C

    Line Integral of a parametric curve

    Homework Statement Evaluate the line integral over the curve C \int_{C}^{}e^xdx where C is the arc of the curve x=y^3 from (-1,-1) to (1,1) Homework Equations \int_{C}^{}f(x,y)ds=\int_{a}^{b}f(x(t),y(t))\sqrt((\frac{dx}{dt})^2+(\frac{dy}{dt})^2)dt The Attempt at a Solution I tried...
  34. T

    Parameterizing Z-Value in Line Integral for Cylinder (Stokes Thm)

    I am a little confused about how to generally go about applying Stokes's Theorem to cylinders, in order to calculate a line integral. If, for example you have a cylinder whose height is about the z axis, I get perfectly well how to parameterize the x and y components, using polar coordinates...
  35. H

    Troubleshooting Line Integrals: Finding Errors in Calculations

    Let $${\tmmathbf\vec{F} = yx^2 \tmmathbf\hat{i} + \sin (\pi y) \tmmathbf\hat{j}}$$, and let $${C}$$ be the curve along the line segment starting at (0,2) and ending at (1,4). $${\int_C \tmmathbf\vec{F} \cdot d \tmmathbf\vec{r} =}$$ My path comes out to be: r=ti +2tj dr=i+2j...
  36. H

    Significance of parabola in a line integral?

    If $${\tmmathbf{r}}$$ and $${\tmmathbf{s}}$$ are piecewise smooth paths, which have the same graph, then they are said to be equivalent paths. They either trace out a set of points in the same direction, or in the opposite direction. If they trace out a curve $${C}$$ in the same...
  37. H

    Setting up a path for a line integral

    If $${C}$$ is the straight line that connects points in the plane $${(x_i, y_i)}$$ and $${(x_f, y_f)}$$, find a path $${\tmmathbf{r} (t)}$$ that traces out $${C}$$ starting at the initial point $${(x_i, y_i)}$$ and ending at $${(x_f, y_f)}$$ as $${t}$$ goes from zero to 1. Now the path that...
  38. S

    How is the Complex Line Integral of a Circle Calculated?

    I read(and numerically calculated) that the line integral around a circle of radius r centered at 0 for the function conjugate(z)/(z - t) is 0 for all t inside the circle. I don't know see how this integration is performed.
  39. S

    Calculating the Line Integral for the Intersection of Two Surfaces

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  40. H

    Line integral to calculate work

    Hi. I have a concrete doubt with this problem. Here's the pic. It asks me to calculate the work done by force P (the ball moves with constant speed). So the solution is in the book and I understood everything, but the problem comes here, the force P in axis y is zero so the work of P should...
  41. G

    Independence of parametrizations of a line integral

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  42. 1

    Another disagreement with course Line integral homework

    Homework Statement http://img39.imageshack.us/img39/6669/wileyplus.png The field is conservative. With their description of C, and using the Fundamental Theorem of Calculus for Line Integrals, would you evaluate f(11/sqrt(2), 11/sqrt(2)) - f(0,0) or f(11/sqrt(2)...
  43. C

    Line Integral - Vector Calculus

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  44. DryRun

    What is the Result of Evaluating a Line Integral Along a Quarter-Circle Curve?

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  45. K

    Line Integral: Meaning of Homotopic

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  46. P

    Line Integral does not match Greens Theorem?

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  47. K

    Line Integral Help: Evaluating F ds on a Curve in 1st Quadrant

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  48. M

    Line Integral Homework: Find Curve & Vector Field for Green's Theorem [Solved]

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  49. A

    Green's Theorem or Simple Line integral Evaluation

    Homework Statement Evaluate the following line integral ∫y^2 dx + x dy where C is the line segment joining the points (-5,-3) to (0,2) and is the arc of the parabola x= 4-y^2 Homework Equations Green's Theorem ∫ Mdx + Ndy = ∫∫ (∂N/∂x - ∂M/∂y ) dy dx The Attempt at a...
  50. A

    How Does a Line Integral of a Scalar Field Differ from a Regular Integral?

    Okay this might be a nooby question, but it bothers me. What is the difference between the line integral of a scalar field and just a regular integral over the scalar field? For a function of one variable i certainly can't see the difference. But then I thought they might be identical in...
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