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}
.

View More On Wikipedia.org
Recent contents Top users
  1. jegues

    Why Does Reversing Limits in a Line Integral Affect the Result?

    Homework Statement See figure attached for problem statement as well as the solution provided. Homework Equations The Attempt at a Solution My only question is why can't I use, 32 \int ^{1}_{2} t\sqrt{8t^{2}+1}dt I realize that this will give me the negative of his answer...
  2. jegues

    Solving Line Integral Problem on Curve z=x+y with Simple Parametric Equations

    Homework Statement Evaluate the line integral, \int_{C}x^{2}yzds, \text{ where C is the curve } z = x + y, \quad x + y + z = 1\text{ from }(1,\frac{-1}{2},\frac{1}{2}) \text{ to } (-3, \frac{7}{2}, \frac{1}{2}) Homework Equations The Attempt at a Solution Here's my attempt at...
  3. A

    Maximum Number of Closed Curves with zero Line Integral

    Hi All, I have been battling with this question for a while. Given a conservative vector field, we know that there are infinitely many closed paths where the line integral evaluated is zero. In fact this is the requirement for a conservative vector field: Every line integral of any closed...
  4. G

    Simple Line Integral becomes troublesome

    I have the parametrization of C x=9cos(t) y=9sin(t) z=-8t 0<=t<=10*pi and I have to calculate \int_c 7x^2+4y^2 -5xy after I transform this I get (7*81cos^2(t) +4*81sin^2(t) -5*81*cos(t)*sin(t))*sqrt(-9sin^2(t)+9cos^2(t)-8) dt from 0 to 10*pi Now that is a monster. What did I do wrong?
  5. K

    Line Integral Fundamental Theorem

    Homework Statement Use Your Phi(from part 1) and the fundamental theorem of line integrals to evaluate the same line integral. (should get the same answer!) The Attempt at a Solution Phi from part 1: Phi = xy+ y^2 +C The line from before go from (0,2) to (-2,0) r(a) = (0,2)...
  6. Char. Limit

    Line Integral along a Parabola

    Homework Statement a. Find a parametric equation to describe a parabola from the point (1,1) to the point (2,4). b. Evaluate the line integral \int_C x ds along the parabolic segment in part a. Homework Equations \int_C x ds = \int_{t1}^{t2} x(t) |r'(t)| dt The Attempt at a Solution Well...
  7. D

    Surface integral to line integral

    I am agonizing about the following integral identity: \frac{d}{dt} \int \int_{g(x,y) \leq t} f(x,y) dx dy = \int_{g(x,y)=t} f(x,y) \frac{1}{\left| \nabla g(x,y) \right|} ds, where ds is the line element. Clearly, using the Heavisite step function, the condition g(x,y) \leq t is...
  8. M

    Line Integral Around Triangle: Curl or Not?

    Homework Statement Without parameterizing the path, determine what the value of the line integral (integral of F dot dr) is, if C is the closed, oriented path that travels around the triangle with vertices (0,0) (5,2), and (-3,6) and F=yi + xj Homework Equations Curl possiblY? The...
  9. J

    Line integral over the perimiter of a hexagram

    Solved Thanks
  10. J

    Line Integral (Flux) Calculation: A(1,4) to B(5,1)

    Homework Statement for \varphi(x,y)=2x+y+10 ,calculate the flux line integral...on a straight line from A(1,4) to B(5,1). Homework Equations The Attempt at a Solution I tried to solve it but didnt get the right answer. first i found the quation of the line which i found to be...
  11. N

    Line integral and continuous gradient

    Homework Statement A table of values of a function f with continuous gradient is given. Find the line integral over C of "gradient F dr" where C has parametric equations x = t2 + 1, y = t3 + t, 0<=t<= 1. Sorry, don't know latex. But here's a picture of the table and values...
  12. C

    Help understanding line integral solution?

    http://img2.imageshack.us/img2/5061/14983795.jpg I have no idea how they simplified the integral to the second step.
  13. C

    Proof of Line Integral Using ∇f & ∇g: R Region, C Curve

    Homework Statement Let f(x,y) and g(x,y) be continuously differentiable real-valued functions in a region R. Show that ∫f ∇g · dr ]= − ∫g ∇f · dr for any closed curve C in R. Homework Equations The Attempt at a Solution I don't really know where to start, so I tried to evaluate...
  14. E

    Solving Line Integral on Curve C

    Homework Statement Evaluate the following line integral on the indicated curve C \int(y^2-x^2)ds C: x = 3t(1+t), y=t^3 ; 0 <= t <= 2 Homework Equations ds = \sqrt{(f'(t))^2+(g'(t))^2}dt The Attempt at a Solution dx/dt = 3+6t dy/dt = 3t^2 ds = \sqrt{(3+6t)^2+(3t^2)^2}dt ds =...
  15. R

    Line integral of a vector field

    Hello, I am attempting to calculate the line integral of the vector field Line integral of a vector field \overline{A}= x^{2} \hat{i} + x y^{2} \hat{j} around a circle of radius R ( x^{2} + y^{2} = R^{2} ) using cartesian coordinates. The appropriate differential line element in cartesian...
  16. R

    Line integral of a vector field

    I am attempting to calculate the line integral of the vector field \overline{A}= x^{2} \hat{i} + x y^{2} \hat{j} around a circle of radius R (x^{2} + y^{2} = R^{2}) using cylindrical coordinates. It is simple enough to convert the x and y components to their cylindrical counterparts, but I am...
  17. X

    Finding Line Integral of Vector Field

    Homework Statement You are given a vector field A= kx2 x. a. First, calculate the line integral of A from x=-2 to x=2 along the x axis. b. Next, calculate the line integral of A between the same 2 points, but use a semicircular path with a center at the origin. Recall that in cylindrical...
  18. N

    Parametrizing Complex Line Integral

    So this is an ultra basic question, but I'm rusty with parametrization techniques and wanted to make sure I was doing this correctly. Let's say I want to evaluate \int_{\gamma} z \: dz where \gamma : [a,b]\rightarrow \mathbb{C} is some path of integration. Now, I figure I can parametrize the...
  19. K

    Integrating a Line Integral over a Parabola: A Challenging Task?

    Homework Statement \int_{C}(xy+\ln{x})\mathrm{d}s where C is the arc of the parabola y=x^2 from (1,1) to (3,9) Homework Equations \int_{C}f(x,y)\mathrm{d}s = \int_{a}^{b}f(x(t),y(t))\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}\mathrm{d}t The Attempt at a Solution Ok, so...
  20. E

    Green's Theorem and Line Integral

    Homework Statement \ointxydx+x^2dy C is the rectangle with vertices (0,0),(0,1),(3,0), and (3,1) Evaluate the integral by two methods: (a) directly and (b) using green's theorem. Homework EquationsThe Attempt at a Solution Evaluating the integral directly: c1: y=0,x=t,dx=dt,dy=o...
  21. Y

    Finding area of ellipse using line integral.

    The standard method of calculating area of ellipse: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 Area = \int_C -ydx \hbox { or } \int_C xdy It is more convient to use polar coordinate x=a cos \theta \; \hbox { and }\; y=b sin \theta dy = b cos \theta \hbox{ Using } \int_C xdy =...
  22. S

    CALC III Line Integral Problem

    Homework Statement [PLAIN]http://img843.imageshack.us/img843/3995/calc3.jpg The Attempt at a Solution Im here asking for some help in direction on how to do these problems so that I can find the solution by myself... I would really really appreciate any help anyone could provide...
  23. K

    Calculating a Line Integral with Triangle C

    Homework Statement hello again, sorry for asking so many questions, i just want to make sure if I am correct or not calculate the line integral y^2dx+x^2dy where line C is the triangle with sides x=1, y=0 and y=x The Attempt at a Solution first of all i tried to find a customization of the...
  24. S

    Greens theorem direction of line integral

    My course notes said that in greens theorem where the closed line integral of F.r = the double integral (...)dxdy the curve c is taken once anti-clockwise, why does it matter which way you take the line integral? Does it matter at all? Thanks
  25. D

    Finding the Correct Parameterization for Line Integral

    Homework Statement I'm attempting Q 3 from ch 16.4 of Stewart (p 1060). We are required to find the line integral where C is the triangle with vertices (0,0), (1,0) and (1,2). The line integral is Int xy dx + x^2*y^3 dy Homework Equations The Attempt at a Solution...
  26. I

    Line integral & logarithmic spiral

    Homework Statement Calcualte the value of \int\limits_L \sqrt{x^2+y^2}dl, where L is an arc of a logarithmic spiral r=ae^{m\phi} between points A(0,a) and B(-\infty,0). Problem: I can't find a value of \phi where x=-\infty or y=a. Homework Equations We parametrise and get...
  27. I

    Solve Real Line Integral: Circle Ranging A(0,R) to B(R,0)

    Homework Statement Calculate the line integral \int\limits_{AB} x^2 dx+ \sqrt{xy}dy , where AB is a part of a circle in the first quarter of carthesian coordinates system ranging from A(0,R) to b(R,0). Homework Equations The Attempt at a Solution Parametrisation of a circle...
  28. M

    Calculating Line Integral of I Using Green's Theorem

    The Integral I is defined by I = Integral F . dr Where F = (x-y, xy) << This is a verticle vector, i just didn'nt know how to write it with latex. And C is a triangle with the vertices (0,0), (1,0) and (1,3) tracked anticlockwise. Calculate the line integral using greens...
  29. W

    Line integral with the inverse square field

    Homework Statement Integrate F(x) = x / |x|^3 along the straight line from (1,0,0) to (2,-2,1). Homework Equations Line integral = int (F dot dx) The Attempt at a Solution I don't know where to start. Usually I do line integrals by parameterizing the line and the vector field...
  30. M

    Line Integral of Vector Field: Is 0 a Meaningful Value?

    Can line integral of a vector field ever be zero? If can, what is the interpretation of this value (0) ? Thanks.
  31. F

    Calculating Work Using Green's Theorem

    Hi everyone. I am going through examples for maths exams and am unsure on the final part of a question I am attempting so hoping you may help me? Homework Statement "Let C be the closed, piecewise smooth curve comprising individual curves C1 and C2 defined by r1 = (x, x2, 1) and r2 =...
  32. H

    Line Integral: Understanding Scalar & Vector

    I want to check my understanding of the line integral: For a scalar line integral, what we have geometrically is the area between a curve a given function, yes? Hence, it can be thought of as a kind of thin wall, correct? And where our function is f(x,y)=1, we have the length of the...
  33. T

    Closed curve line integral of gradient using Green's Theorem

    Apostol page 386, problem 5 Homework Statement Given f,g continuously differentiable on open connected S in the plane, show \oint_C{f\nabla g\cdot d\alpha}=-\oint_C{g\nabla f\cdot d\alpha} for any piecewise Jordan curve C. Homework Equations 1. Green's Theorem 2. \frac{\partial...
  34. J

    How Do You Evaluate a Line Integral Along a Parametric Curve?

    So, my multivariable class has just started line integrals, and I could use a little help with them. The problem I'm currently working on says: Evaluate the line integral, where C is the given curve: \int\limits_C \! xy \,ds C: x=t^2, y=2t, 0 \le t \le 1 I realize that, by eliminating the...
  35. J

    Line Integral of ydx +zdy + xdz on the Intersection of Two Curves

    Homework Statement Compute the line integral of \intc ydx +zdy + xdz where c is the intersection of x^2 +y^2+z^2= 2(x+y) and x+y=2 (in the direction clockwise as viewed from the origin) Homework Equations The Attempt at a Solution While attempting this problem I had a few...
  36. N

    Line Integral vs. Surface Integral: Range of t?

    what is the different between line integral and surface integral? If we parameterize curve by x=t , y=t , what is the range of t ? Is it 0<= t <=1? why?
  37. A

    Scalar potential and line integral of a vector field

    Homework Statement Homework Equations Given above. The Attempt at a Solution I attempted this problem first without looking at the hint. I've defined F(r) as (B+A)/2 + t(B-A)/2, with dr as (B-A)/2 dt . Thus F(r)dr = ((B+A)/2)*((B-A)/2)+((B-A)/2)^2 dt When I integrate this from -1 to 1 I...
  38. B

    Surface Integral - or Line Integral?

    Homework Statement Air is flowing with a speed of 0.4m/s in the direction of the vector (-1, -1, 1). Calculate the volume of air flowing per second through the loop which consists of straight lines joining, in turn, the following (1,1,0), (1,0,0), (0,0,0), (0,1,1), (1,1,1) and (1,1,0)...
  39. D

    Line integral, vector calculus

    Evaluate the line integral Force field is the integral in the form of integrand ( (2x dx+ 2y dy + 2 zdz)/r^2). the domain of integral is C, C = C1 + C2. C1 is the line segment from (1; 2; 5) to (2; 3; 3). C2,arc of the circle with radius 2 and centre (2; 3; 1) in the plane x = 2. The...
  40. A

    Line integral uncertain about direction.

    Homework Statement Evaluate \int[(3x-y)dx-xdy] where C consist of the parabola y=x^2 from (0,0) to (1,1) and then the line segment from (1,1) to (0,1) Homework Equations The Attempt at a Solution i did the integral of the y=x^2 parametrized x=t y=t^2 from 0 to 1 then i got my 1/2 but for the...
  41. I

    Is the Electric Field Always Conservative or Can it be Non-Conservative?

    In some books I have seen: \oint \mathbf{E} \cdot d\mathbf{s}=0 Since the Electric Field is meant to be conservative. Elsewhere, however, I have also seen: \oint \mathbf{E} \cdot d\mathbf{s} = -\frac{d\Phi_B}{dt} What's going on here? Thanks
  42. L

    Is the ordinary integral a special case of the line integral?

    Can I consider the ordinary integral over the real line a special case of the line integral, where the line is straight and the field is defined only along the line?
  43. C

    Line integral with respect to arc length

    In a line integral with respect to arc length, we have something like f(x, y)ds "inside" the integral sign. The ds tells us that we are working with the arc length function s, taking diferences (s_K+1 - s_k) in the sums that tend to the line integral. Question: do we shall understand that...
  44. S

    Line integral around an ellipse

    Homework Statement What is \int_{\gamma} xy dx + x^2 dy in each of the following cases? 1. \gamma is the lower half of the curve 2x^2 + 3y^2 = 8, traveled from (2,0) to (-2,0). 2. \gamma is the full curve 2x^2 + 3y^2 = 8, traveled counterclockwise. Homework Equations The line...
  45. P

    Line integral in polar/spherical system?

    Hello, sorry for my English;D Homework Statement Can a vector field exist in polar/spherical system? is it possible to define line integral in these systems? does it make any sense a vector field defined in polar system, ex. \vec A\left(r,\varphi\right)=r^3? and a line integral from...
  46. R

    Stokes theorem and line integral

    Homework Statement Prove that 2A=\oint \vec{r}\times d\vec{r} Homework Equations The Attempt at a Solution From stokes theorem we have \oint d\vec{r}\times \vec{r}=\int _{s}(d\vec{s}\times \nabla)\times \vec{r}= \int _{s}(2ds\frac{\partial f}{\partial x},-ds+ds\frac{\partial...
  47. O

    How do i maximize the line integral?

    suppose i have a nonconservative vector field. and there is a path going from point A to point B. How do i determine the path taken from A to B such that the line integral is maximized? edit: actually after thinkin about it, this might be an undefined problem unless there is some constraint on...
  48. P

    Calculating Line Integral of C from (1,0) to (3,1)

    Homework Statement Suppose C is the line segment from the point (1,0) to the point (3,1). Compute the line integral intC {( xdx + (x + y)}dy Homework Equations The Attempt at a Solution i graphed the line that connects(1,0) to (1,3) and i got the equation of that line so y...
  49. R

    Line Integral: Solving for \ointr.dr=0

    Homework Statement What is the result of this? \ointr.dr=? Homework Equations The Attempt at a Solution \ointr.dr =\ointrdr=\int^{a}_{a}rdr = \frac{r^2}{2} \left| ^{a}_{a} = 0 Is it correct?
  50. S

    Line integral of a conservative vector field

    Homework Statement This is an example in my book, and this is the information in the question. Find the work done by thr force field F(x,y) = (1/2)xy[B] i + (1/4)x^2 j (with i and j vectors) on a particle that moves from (0,0) to (1,1) along each path (graph shows a x=y^2 curve from (0,0)...
Back
Top