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
  1. I

    Prove a Statement about the Line Integral of a Vector Field

    Homework Statement Given the vector field \vec{v} = (-y\hat{x} + x\hat{y})/(x^2+y^2) Show that \oint \vec{dl}\cdot\vec{v} = 2\pi\oint dl for any closed path, where dl is the line integral around the path.Homework Equations Stokes' Theorem: \oint_{\delta R} \vec{dl}\cdot\vec{v} = \int_R...
  2. A

    Integrating a Line Integral with Parametric Equations

    Homework Statement Evaluate the line integral ∫c y2 dx + 2xy dy, where C, is the path from (1, 2) to (2, 4) parametrised by r(t) = (t2 + 1)i + (2t2 + 2)j , 0 ≤ t ≤ 1 Homework Equations I worked out the velocity magnitude |v(t)| as 2t√5 The Attempt at a Solution I simply integrated...
  3. DryRun

    What is the method for evaluating a line integral with a given parametrization?

    Homework Statement http://s2.ipicture.ru/uploads/20120204/iuPLuS1l.png The attempt at a solution x=t^2-1 and y=t^2+1 \frac{dx}{dt}=2t and \frac{dy}{dt}=2t The line integral is of the form: \int P\,.dx+Q\,.dy So, i use direct substitution: \int^1_0 4t^3-2t\sin(t^2-1)+2t\cos(t^2+1)\,.dt Is...
  4. I

    How to visualize a line integral

    Hi, I am trying to get a useful heuristic picture of a line integral, like the area under a curve for an ordinary integral. My current one is: if I place a particle in a force field, then the line integral from point A to B is the change in kinetic energy of the particle from A to B. This...
  5. DryRun

    Independence of path of line integral

    Homework Statement http://s2.ipicture.ru/uploads/20120117/ReWSCD1f.jpg The attempt at a solution \frac{\partial P}{\partial y}=\frac{2y}{x^3} \frac{\partial Q}{\partial x}=\frac{2y}{x^3} \frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y} According to my notes: Both functions are...
  6. J

    Is this line integral correct?

    Homework Statement ∫C1(0) dz / (z * sin2(z)) Homework Equations Residue Theorem material The Attempt at a Solution z * sin2(z) = z * (1/2 - cos(2z)/2) = z * [1/2 - (1/2)∑(-1)n(2z)n/(2n)! ] = z3 + ... ---> z * sin2(z) has a zero of order 3 at z = 0 ---> 1/(z * sin2(z))...
  7. B

    How Is Area Calculated Using a Line Integral for an Ellipse?

    Homework Statement Find the area swpet out by the line from the origin to the ellipse x=cos t and y=sin t as t varies from 0 to t_0 where t_0 is constant between 0 and 2 pi Homework Equations Use A= \frac{1}{2} \oint_{C} y dx -xdy The Attempt at a Solution How does one...
  8. J

    Seemingly easy complex line integral

    Homework Statement Find ∫(ez+cos(z))/z dz integrated over C1(0) Homework Equations Theorem 6.10 (Cauchy's integral formula) Let f be analytic in the simply connected domain D and let C be a simple closed positively oriented contour that lies in D. If z0 is a point that lies interior to C...
  9. P

    Line Integral That's Not Working Out

    Let \alpha be circle in the complex plane centered at z=1 with radius r=3/2. I proceed by partial fraction decomposition and then use Cauchy's Integral Formula. \int_\alpha \frac{z^7 -1}{z^6 - z^2}dz = \int_\alpha zdz - \int_\alpha \frac{1}{z^2}dz +\frac{i}{2}\int_\alpha \frac{z-i}{z^2 - i}dz...
  10. A

    Line integral, Error in my textbook?.

    Homework Statement Evaluate the line integral: int(ydx+xdy) where the path C is y=sinx from (0,0) to (pi/2,0) Homework Equations The Attempt at a Solution (pi/2,0) is not a solution to y=sinx. I could use the fundamental theorem but for my potential function I get F(x,y)=xy...
  11. B

    Compute Line Integral: (x/y) from (2,4) to (10,100)

    The question is compute the integral over c of (x/y) where c is the line segment from (2,4) to (5, 25) followed by the parabolic arc from (5, 25) to (10, 100) I tried setting this up in terms of x and then y using the line integral formula but I am got a negative answer which i know can't be...
  12. B

    How do I evaluate this Line Integral over the boundary of a unit disk?

    Homework Statement Let C be the (positively oriented) boundary of the first quadrant of the unit disk. Use the definition of the line integral to find ∫(xy)dx+(x+y)dy Homework Equations x=rcos(x) y=rsin(x) dx=-sin(x) dy=cos(y) 0≤ t ≤ ∏/2 The Attempt at a Solution...
  13. B

    Use the defiinition of a line integral to evaluate

    ∫Homework Statement Use the definition to find the line integral of F(x,y) = (y,x) along each of the following paths. The parabola y = x^2 from (-1,1) to (1,1) Homework Equations F(x) = gradientf(x) ∫F(x) dx = f(b) - f(a) The Attempt at a Solution I tried (y,x) dot...
  14. M

    What is the line integral for the given line segments and parametric equations?

    Homework Statement Find the line integral of ∫ x+yz dx + 2x dy + xyz dz C consists of line segments from (1,0,1) to (2,3,1) and from (2,3,1) to (2,5,2). Homework Equations r=(1-t)<r0> + t<r1> 0<t<1 The Attempt at a Solution I split up the two line segments into C1 and C2...
  15. J

    Complex Line Integral (not too hard)

    Homework Statement ∫dz/(z4+1) integrated over the curve C1(1+i) Homework Equations The only thing we learned in this chapter is Cauchy's integral formula, so I'm assuming that comes in somehow. The Attempt at a Solution ∫dz/(z4+1) = ∫dz/(z+1+i)(z+1-i)(z-1+i)(z-1-i) Not bad...
  16. L

    Mass of 2 dimensional object - line integral

    Homework Statement Find the mass and the coordinates for the center of mass of a thin wire formed like a quarter circle. Homework Equations Circle equation: x2+y2=r2 Mass density: rho=x+y The Attempt at a Solution I know that: x2+y2=cos2(t)+sin2(t)=1 This leads to...
  17. J

    Complex Line Integral (should be easy)

    Homework Statement Using a partial fraction decomposition, show that if z lies in the right half plane and C is the line segment from 0 to z, then ∫C dz/(z2+1) = i/2 Log(z+i) - i/2 Log(z-i) + π/2 Homework Equations Log(z) = ln(z) + i Arg(z) (maybe relevant?) The Attempt at...
  18. D

    Computing the line integral of the scalar function over the curve

    Homework Statement f(x,y) = \sqrt{1+9xy}, y = x^{3} for 0≤x≤1 Homework Equations The Attempt at a Solution I don't even know how to start this problem. I thought about c(t) since that's all I have been doing, but there isn't even c(t). I only recognize domain. Can anyone help me...
  19. A

    Find magnetic force on semicircle using line integral

    Here is the question: A very thin wire which follows a semicircular curve C of radius R,lies in the upper half of the x-y plane with its center atthe origin. There is a constant current I flowing counter clockwise, starting upward from the end of the wire on the positive x-axis and ending...
  20. B

    Evaluate the Following Line Integral Part 3

    Homework Statement \displaystyle \int x^2dx+y^2dy+z^2dz where C is the line segment from(0,0,0) to (1,2,-1) and (1,2,-1) to (3,2,0) Homework Equations \displaystyle\int_c \vec F(t) d \vec r(t)= (x^2 i+ y^2 j+z^2 k)d \vec r(t) where d \vec r(t) for C_1=ti+2tj-tk and d \vec r(t) for...
  21. B

    Evaluate the Following Line Integral Part 2

    Homework Statement \displaystyle \int_c zdx+xdy+ydz where C is given by t^2\vec i +t^3 \vec j +t^2 \vec k Can this \displaystyle \int_c zdx+xdy+ydz be written as \displaystyle \int_c z\vec i+x \vec j+y \vec k? I believe I need to evalute the integral \displaystyle \int_c \vec F(...
  22. B

    Evaluate the Following Line Integral Part 1

    Homework Statement \int_{c}cos (x)dx+sin(y)dy where c consist of the top half of the circle x^2+y^2=1 from (1,0) to (-1,0) The Attempt at a Solution Do I parameterise x=t and then y becomes y= (1-t^2)^{1/2}...? Replace the corresponding dx and dy and then integrate between the limits?
  23. A

    Simplifying Complicated Trigonometric Integrals

    Homework Statement \int_{C}|y|ds where C is the curve (x^{2}+y^{2})^{2}=2^{2}(x^{2}-y^{2}) Homework Equations The Attempt at a Solution i used polar coordinates x = r cos \theta and y = r sin \theta then substituted into the equation to get r = 2\sqrt{cos 2\theta} since r\geq0 gives...
  24. W

    Measuring the Length of a Parabolic Path with Line Integral

    Hi experts what is line integral - for example if I can draw graph of parabola and i can calculate the area under the graph. But how can i measure the length of parabolic path.
  25. C

    Line integral to determine area of sphere?

    Find the area of the surface consisting of the part of the sphere of radius 2 centered at origin that lies above the horizontal plane z = 1. (Equation of this sphere is given by x^2 + y^2 + z^2 = 2^2 .) x^2+y^2+1=4 x^2+y^2=3 This is the base of the solid. But how do we find the required...
  26. N

    How Do You Compute Work Done in a Vector Field with Polar Coordinates?

    Homework Statement The Problem states: Given the force vector field(in polar coordinates) : F(r,\theta)=-4Sin\thetai+4Sin\theta j, compute the work done in moving a particle from (1,0) to the origin along the curve whose polar equation is : r=e^{-\theta} The Attempt at a Solution I...
  27. H

    Parametrizing a Line Integral: Finding the Easiest Approach

    How do you work out the parameterization for a line integral. I have this example, and the closed curve C bounds the lines y=0, x=2 and y^2 = 8x. In the solution to the problem it states that there are many parameterizations available. So I just wanted to know, how do you work out the...
  28. B

    What is the value of this complex line integral?

    find the value, \int\limits_{0}^{2\Pi} e^{-\sin t} \sin\lbrace (\cos t ) - (n-1) t \rbrace dt ? I have no idea...
  29. B

    Complex Line Integral Value for Natural Numbers n=1,2,3..

    I can't find the value, for natural number n = 1, 2, 3, ... I = \int\limits_{C}\dfrac{e^{iz}}{z^n} dz find the value. where z(t) =e^{it} , 0\leq t \leq 2\Pi
  30. S

    Calculate Line Integral of Vector Field f(x,y) over Curve C | Homework Problem

    Homework Statement calculate the integral f · dr for the given vector field f(x, y) and curve C: f(x, y) = (x^2 + y^2) i; C : x = 2 + cos t, y = sin t, 0 ≤ t ≤ 2π (2pi) Homework Equations Would the vector F simply be <(x^2+y^2),0> since there is no j component? The solution is 4pi...
  31. T

    Line integral with vector function on circular path.

    I'm not getting the answer from the back of the book for some reason. Is the book wrong or am I wrong? Homework Statement calculate \intf · dr for the given vector field f(x, y) and curve C: f(x, y) = (x^2 + y^2) i; C : x = 2 + cos t, y = sin t, 0 ≤ t ≤ 2πHomework Equations itex]\int[/itex]f ·...
  32. J

    Line integral of sin cos function

    Homework Statement integral (sin^4(x) + cos^4(x))^.5 dx Homework Equations sin^2(x) = (1 - cos^2(x)) The Attempt at a Solution cos^4(x) + {1 - cos^2(x)}^2 = 2cos^4(x) -2cos^2(x) + 1 subst u = cos^2(x) integral (2u^2 - 2u +1)^.5
  33. Y

    Line Integral w/ Respect to Arc Length & x/y

    There are line integral with respect to arc length and line integral with respect x/y. I know \int_C Pdx+Qdy is useful to calculate the work. When do we need the line integral with respect to arc length?
  34. A

    Line Integral, Green's Theorem

    Homework Statement \int_{C} (xy^{2}-3y)dx + x^{2}y dy G is finite region enclosed by: y=x^{2} y=4 C is boundary curve of G. Verify Green's Theorem by evaluating double integral and line integral. The attempt at a solution Q = x^{2}y dQ/dx = 2xy P = xy^{2}-3y dP/dy =...
  35. Rasalhague

    Bachman's line integral versus classical line integral

    Bachman's "line integral" versus "classical line integral" David Bachman A Geometric Approach to Differential Forms http://arxiv.org/abs/math/0306194 When Bachman talks, in Appendix A, about "classical" line, surface, volume integrals, does he mean integrals of differential 0-forms (scalar...
  36. O

    Integrating a Line Integral Along a Curve with Given Boundaries

    Homework Statement ∫(zdx+xdy+ydz) along the curve C: x(t)= cos(t), y(t)= sin(t), z = 3t, Boundaries are 0 and 2pi Homework Equations General integration and differentation. The Attempt at a Solution given the values I calculated that: Using chain rule: dx = -sintdt...
  37. S

    Line Integral of a Vector Field over a Half Sphere using Stoke's Theorem

    Homework Statement F = ( 2y i + 3x J + z2 k where S is the upper half of the sphere x2 + y2 + z2 = 9 and C is its boundary. Homework Equations The Attempt at a Solution I used Stoke's Theorem and found the solution to be 36 pi, but when I use line integral to verify, using...
  38. F

    What is with this line integral?

    Homework Statement http://img534.imageshack.us/img534/6859/unledei.jpg 3. The Solution [PLAIN][PLAIN]http://img607.imageshack.us/img607/3104/unledfe.jpg Why did they switch the order of x - 2xy3 in Green's Theorem?
  39. K

    Calculating Area of Intersected Cylinders Using Line Integral

    Homework Statement Two circular cylinders of radius a intersect so that their axes meet at right angles. Use a line integral to find the area of the part from one cut off by the other. Homework Equations line integral formula The Attempt at a Solution I'm lost as to where to set...
  40. Z

    Line Integral and Vector Field Problem

    Homework Statement Find the work done by the force field F(x,y) = x sin(y)i + yj on a particle that moves along on the parabola y = x^2 from (-1,1) to (2,4). Homework Equations Work = line integral of the dot product of Field vector and change in the path The path is parabola equation...
  41. Z

    Finding the mass and center of mass of a wire using a line integral.

    Homework Statement Find the mass and center of mass of a wire in the shape of the helix x=t, y=\cos{t}, z = \sin{t}, 0 \le t \le 2 \pi, if the density at any point is equal to the square of the distance from the origin. Homework Equations Arc length formula: ds =...
  42. R

    Another complex line integral question

    I have to integrate |z|2dz from 0 to 1 + 2i using the indicated paths. The first path is a straight line from the origin to 1 + 2i and the second has two lines, the first going from 0 to 2i along the y-axis and then from 2i to 1 + 2i, a line parallel to the x axis. For the first path, the...
  43. R

    Line integral of complex function

    I have to evaluate this line integral in the complex plane by direct integration, not using Cauchy's integral theorems, although if I see if a theorem applies, I can use it to check. \int (z^2 - z) dz between i + 1 and 0 a) along the line y=x b) along the broken line x=0 from 0 to 1...
  44. J

    How to compute this line integral

    Hi guys, can anyone help me with evaluating this: \int^{}_C |y| \,ds where C is the curve (x^2+y^2)^2=r^2(x^2-y^2) any hints with the parametrization?
  45. L

    Line Integral Question (Vertical line issues)

    Homework Statement A wire lies along the piecewise linear curve extending from the point (2,2) to the point (12,4) to the point (12,9). If the density of the wire is given by (xy)=2xy+6x, use a line integral to find the mass of the wire. Homework Equations The Attempt at a Solution...
  46. C

    Line Integral: Evaluating Along Circular Path from P1 to P2

    Homework Statement Evaluate the line integral along the segment P1(0,3) to P2(-3,0) of the circular path shown in figure. Figure basically shows a circle with a radius of 3. The part that i have to evaluate is from the y-axis (P1) to the x-axis (P2), basically a quarter of the circle...
  47. B

    Work of a Line integral correction please

    Homework Statement Compute the Work of the following line integrals in the vector field \vec{V}=(2x^{2}-3y;4xy;3x^{2}z) Homework Equations For the following lines: Curve1: \vec{r}(a)=(a,a,a^{2}); \ 0\le a \le 1 Curve2: \vec{r}(a)=(a,a^{2},a); \ 0\le a \le 1 The Attempt at a Solution...
  48. N

    Fundamental Theorem of Line Integral question

    The question is suppose that F is an inverse square force field, that is, F(r)=cr/|r^3| where c is some constant. r = xi + yj + zk. Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d2 from those points to the origin...
  49. M

    Line Integral on R2 Curve in Polar Coordinates

    Homework Statement Consider a curve in R2 given in polar coordinates r=r(θ) for θ1<= θ <= θ2. Show that the line integral is equal to the integral from θ1 to θ2 of f(r*cosθ, r*sinθ) sqrt (r^2 + (dr/dθ)^2) dθ Homework Equations x= cos θ, y= sin θ The Attempt at a...
Back
Top