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. tony873004

    Evaluate the line integral by two methods

    Homework Statement Evaluate the line integral by two methods: (a) directly and (b) using Green's Theorem. \oint_C {xy^2 \,dx\, + \,x^3 \,dy} . C is the rectangle with vertices (0,0), (2,0), (2,3) and (0,3). The Attempt at a Solution I get the correct answer using Green's theorem. But I...
  2. F

    Evaluating Line Integral: Curl of F and its Relation

    1. Evaluate the line integral∫F . dr with F = 3(-y,x,0) from (a,0,0) to (a,0,2πb) along a straight line. 2. Do the same along a circular helix between the two points, parameterised as r = (a cosλ, a sinλ, bλ) 3. Compute the curl of F. How does this relate to the two integral calculations...
  3. C

    Parabolic Cylinder line integral

    1.Homework Statement (Parabolic Cylinder) find the area of the surface extending upward form x^2 + y^2 =1 to z = 1 - x^2 using line integral 2. Could some one please outline the method to solving this. I tryed using spherical corridinates but am unsure if this was correcect The...
  4. D

    Line Integral: Applying Standard Test to Find Dependency

    Hiya, I've just done a line integral question, and the final part of the question is, "by applying a standard test, determine whether the value of the line integral depends on the path followed between the given initial and final points." The only standard test, I know, is dq/dx, and dp/dy to...
  5. C

    Curl about an elipse. Line integral of vector field

    Homework Statement It can be shown that the line integral of F = xj around a closed curve in the xy - plane, oriented as in Green's Theorem, measures the area of the region enclosed by the curve. (You should verify this.) Use this result to calculate the area within the region of the...
  6. E

    Why Is the Line Integral of a Parallel Vector Field Zero on a Sphere?

    Homework Statement If the vector field F(x,y,z) is everywhere parallel to R and C is a curve drawn on a sphere with the center at the origin, then: \int\stackrel{C}{} F . dR = 0 why? The Attempt at a Solution Im not exactly sure if I understand the problem, but this is the explanation...
  7. C

    Magnetic Field Line Integral Problem

    Homework Statement What is the line integral of B between points i and f in the figure? Homework Equations Ampere's Law: ∫B∙dl = Ienclosed * μ0 note: the integral on the left is a line integral. The Attempt at a Solution I applied Ampere's Law. I know that the contribution...
  8. I

    How Can I Correctly Solve a Line Integral for Atmospheric Scattering?

    Hello! I am working on atmospheric scattering and therefore I have to calculate the optical depth along a given path through the atmosphere. The integral I have to solve is this one: \int_{P_a}^{P_b} f ds \quad with \quad f:R\rightarrow R, h\rightarrow e^{(-h/H_0)} P_a und P_b are...
  9. C

    Computing a Line Integral: Stokes' Thm

    Homework Statement Compute the line integral of v = 6i + yz^2j + (3y + z)k along the path (0,0,0) -> (0,1,0) -> (0,0,2) -> (0,0,0). Check your answer using Stokes' Thm Homework Equations The Attempt at a Solution I've tried breaking into three pieces. The first with dx = dz =...
  10. D

    Why Are the Terms Squared After Substitution in Green's Theorem Integral?

    i would like to find the area bounded by the curve (((x^2)/(a^2))+((y^2)/(b^2)))=xy/(c^2) i used the substitution given x=(ar)cos(theta) and y=(ar)sin(theta) i get : (r^2cos^2(theta)+r^2sin^2(theta))^2=xy/(c^2) thus r^4=xy/(c^2) substituting x=(ar)cos(theta) and...
  11. M

    Line Integral Problem: Evaluating F(x,y) on Lower Half of Unit Circle

    Homework Statement Let F=x^{2}i+2xyj, and let C be the lower half of the unit circle, with perametrization r(t)=<cos(t),sin(t)>,\pi\leqt\leq\pi. Evaluate \ointF\cdotdr. Homework Equations The Attempt at a Solution The first thing I tried to do was to find a function f(x,y) so...
  12. H

    Simple parametrized line integral

    [SOLVED] Simple parametrized line integral This is from an example in my textbook. They want you to evaluate the line integral: \int_{C} y dx + 2x dy for the straight line segment in the plane from A(1, 1) to B(2, 4). The example says that this segment can be parametrized as x = 1 +...
  13. J

    Multivariable Calculus: Force along Line Integral

    Homework Statement Part 1: A 160lb man carries a 25lb paint can up a spiral staircase, which has radius 20 feet, completes 3 revolutions, and has final height 90 feet. What is the work done? Part 2: This time, the man's paint can leaks at a constant rate such that he loses 9lbs of paint...
  14. T

    How Do You Calculate a Line Integral in Physics for a Given Parametric Curve?

    Homework Statement Let x = \cos^{3} t and y = \sin^{3}t ( 0 \leq t \leq 2 \pi ). Also \rho(x,y) = k . Find I_0 = \int_{C} (x^{2} + y^{2}) \ dm Homework Equations The Attempt at a Solution So m = \int_{C} k \ ds = 3k \int_{0}^{2 \pi} \cos t \sin t \ dt . Then dm = 3k \cos t \sin t \...
  15. B

    Vector calculus - line integral computation

    Compute the line integral \int_{C} F\cdot dr where F = -y i + x j. The directed path C in the xy-plane consists of two parts: i) a left semicircle from (0, -1) to (0, 1) with center at the origin, and ii) a straight line segment from (0,1) to (2,1). i) r(t) = cos t i + sin t j [pi/2 <=t<=...
  16. B

    Vector calculus - line integral

    Suppose that F is an inverse square force field; this is, F(r) = cr/ |r|^{3} for some constant c, where r = xi + yj + zmbfk. 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 these points to the origin. Not exactly...
  17. E

    How Do You Calculate a 3D Line Integral Along Multiple Paths?

    Dear Users Please help me in starting this problem I have tried my best but all in vain Calculate line integral v=X^2{x(Cap)}+2yz{y(Cap)}+y^2{z(Cap)} from origion to point (1,1,1) by three different routes (a) (0,0,0)→(1,0,0)→(1,1,0)→(1,1,1) Now there are three parts in this problem.I want...
  18. O

    Evaluating Line Integral for xy Plane Semi-Circle

    Homework Statement This comes from Mathematical Methods for the Physicist from Susan Lea, Chapter 1 Question 25 Part B (Incase anyone is familar with the book). The question asks to evaluate the line integral integral (u*dl) where the vector u is: u = x*y^2 i + y*x^2 j along the...
  19. T

    Elliptic Line Integral: Solving for Circulation Around an Ellipse

    Homework Statement Let C be the ellipse with center (0,0), major axis of length 2a, and minor axis of length 2b. Evaluate \oint_C xdy - ydx.Homework Equations I solved this two ways. First I parameterized x and y as x=a \cos \theta and similarly for y. I also applied Green's theorem, which...
  20. E

    Line integral with respect to x or y

    I am confused about how \int_C f(x,y) dx = \lim_{||P|| \to 0} \sum_{i = 1}^n f(x_i^*,y_j^*) \Delta x_i is different from \int f(x,y) dx where P is a partition and its norm is the length of its largest elements. The index i represents an element in that partition and the asterik means...
  21. J

    How Does Orientation Direction Affect Stokes' Theorem Calculations?

    Homework Statement F = <x^2 y^3 z, sin(xyz), xyz> S is part of the cone y^2 = x^2 + z^2 that lies between y = 0 and y = 3. Oriented in the direction of the positive y-axis. Homework Equations The Attempt at a Solution I know how to do the integral, and I get the correct answer except it's the...
  22. M

    Solve a very complex line integral

    Homework Statement Solve I = \int_{\gamma} f(z) dz where \gamma(t) = e^{i \cdot t} and 0 \leq t \leq \pi Homework Equations Do I use integration by substitution?? The Attempt at a Solution If I treat this as a line-integral I get: I = \int_{a}^{b} f(\gamma(t)) \cdot \gamma'(t)...
  23. R

    Line Integral - Work Done by Vector Field Along Curve

    Homework Statement A curve is formed by the intersection of y^2/9 + z^2/4 = 1 and the plane x-2y-3z = 0. The particle moving along the curve goes from (6,0,2) to (-6,0,-2). Find the work done on it by the vector field F(x,y,z) = -yi + xj + yzk. Homework Equations I'm going to need to...
  24. K

    Line Integral: Solving the equation (2xe^y)dx + (x^2e^y)dy from (0,0) to (1,-1)

    Int ((2xe^y)dx + (x^2e^y) dy) from (0,0) to (1,-1) I get the answer 2/e, while my book says 1/e. Am I right or wrong?
  25. M

    Line integral problem, lost when they sub, have work written out

    Hello everyone~ I have the following problem, its done in the book but I'm lost on how they came to the final answer. Evaluate the line integral: Integral over C xy dx + (x-y) dy, C is the line segment from (0,0) to (2,0) and (2,0) to (3,2). C = C1 + C2; On C1: x = x, y = 0; dy...
  26. M

    Line integral question, answer is here, just confused on it

    Hello everyone I'm confused on this line integral. The substiution is easy but I'm not sure where 2t is coming from... integral over C x^2*y*sqrt(z) dz; C: x = t^3; y = t; z = t^2; 0 <= t <= 1 integral over C x^2*y*sqrt(z) dz = integral 0 to 1 (t^3)^2 (t) sqrt(t) * 2t dt =...
  27. M

    Line Integral + Green Theorem problems

    Question 1) Integrate y^2 dx + zx dy + dz, along the circle x^2 + y^2 – 2y = 0 to (1, 1, 0) I am not sure how to begin on this problem. Would it be beneficial to convert to polar coordinates? The thing that is throwing me off is the fact that the circle’s equation is not the normal circle...
  28. J

    Line Integral Problems on Ellipse Boundary | Compute Integrals Directly

    Homework Statement This is my problem: Compute the following three line integrals directly around the boundary C of the part R of the interior ellipse (x^2/a^2)+(y^2/b^2)=1 where a>0 and b>0 that lies in the first quadrant: (a) integral(xdy-ydx) (b) integral((x^2)dy) (c) integral((y^2)dx)...
  29. W

    Work Problem, Line Integral Fun (Calc 3)

    Homework Statement Find the work done by the force field F(x,y) = x*sin(y) i-hat + y j-hat on a particle that moves along the parabola y = x^2 from (-1,1) to (2,4). 2. The attempt at a solution Please see attached file. The answer I get seems really, really, complicated and I have a...
  30. O

    Line integral and path dependence question

    Given F = iy - jx (this is my first post; not sure how you do vector notation here but I'm showing vectors in bold - hope that works). The problem is to show that this is a non-conservative force by integrating from the origin to (1,1) (ie, the path is y=x), and then do it again from the origin...
  31. F

    Efficient Line Integral Computation on Cartesian Coordinates

    In my emag course we are reviewing vector calculus. I've forgotton a lot over the summer, so I just want to make sure I'm doing this properly. question) \vec E = \hat x y + \hat y x Evaluate \int \vec E \cdot d\vec l from P_1(2,1,-1) to P_2(8,2,-1) along the parabola x = 2y^2 . sol)...
  32. A

    Line integral w/ constants only only in integrand

    Hello, I want help in the line integration of: Integral( 1 dy + 3 dx ), over the curve C. Where C is the union of two line segments: Line 1 from point (0,0) to (1, -3) Line 2 from point (1, -3) to (2,0) The thing is I do not know what to do with the integrand being composed of...
  33. S

    How Do Line Integrals Relate to Magnetism?

    Line Integral and Magnetism Help! Here is a problem given to me as a takehome quiz I am having trouble even starting being that I missed that day we went over line integrals. Any hints or tips are much appreciated. http://ez-files.net/download.php?file=4ec07c5df1e8223123523fb11f038d39"
  34. U

    What is the value of the line integral

    Suppose F=F(x,y,z) is a gradient field with F=\nabla f, S is a level surface of f, and C is a curve on S. What is the value of the line integral \int_C F dr? I know the answer is 0 but I cannot visualize why it would be zero?
  35. denian

    Line Integral Question on Mr. Nerdy Forum - Get Answers Now!

    http://www.mrnerdy.com/forum_img/ask.JPG is the first stepc orrect? TQ.
  36. M

    How Do You Solve the Line Integral on a Semi-circle and Line Segment?

    I have been working on the following line integral: \int_{T}^- {(-x^2y)dx + (y^2x)dy} where T is the closed curve consisting of the semi-circle x^2 + y^2 = a^2 (y>0) and the segment (-a,a) I will tackle this in two steps: 1) solve x^2 + y^2 = a^2 (y>0) for y and substitute...
  37. L

    Line Integral: Int(yzdy) but

    Hi all, (This is part of a DJGriffiths, 3rd ed., problem: Prob. 1.28) Line Integral: Int(yzdy) [lower limit = (1,1,0); upper limit = (1,1,1)] but y does not change and is supposed to be integrated, while z changes and is not integrated. I have 2 questions: [1] I take z out of the integral...
  38. B

    Line integral and vector fields

    Hi, I'm having trouble with the following question. Q. Let p be a real constant and \mathop F\limits^ \to = \left( {yz^p ,x^p z,xy^p } \right) be a vector field. For what value of p is the line integral \int\limits_{C_2 }^{} {\mathop F\limits^ \to \bullet d\mathop s\limits^ \to } =...
  39. B

    Line integral examples in book

    Hi, I've just started working on line integrals and I don't understand one of the examples in my book. \int\limits_C {y^2 dx + xdy} Where C is the arc of the parabola x = 4 - y^2 from (-5,-3) to (0,2). The book proceeds by suggesting that y is taken as the parameter so that the arc C...
  40. T

    Line integral and parametrization

    I know this is dumb question but for some reason I have not been able to get the right answer to the following problem: \int_{c} 2xyzdx+x^2 zdy+x^2 ydz where C is a curve connecting (1, 1, 1) to (1, 2, 4). My parametrization is (1, 1+t, 1+3t). My limits are the problem...I think. By...
  41. A

    Why Is My Line Integral Calculation Incorrect?

    Calculate the line integral of the function v=(y的平方, 2x(y+1), 0) from the point a=(1,0,0) to the point b=(2,2,0) The correct Ans: 11 However, when I was calculuating the problem, I supposed(is that the right word?) to make the parameter equations, x=1+t y=1+t z=0 where 0<t<1...
  42. A

    What is the geometric meaning of a line integral?

    I understand how to evaluate a line integral, but I don't know what it represents geometrically. Say you have \int_{C} x^4yd\mathbf{s}. What does this mean geometrically? I can see that \int d\mathbf{s} is the length of the arc (am I correct?), but I just can't seem to figure out what...
  43. H

    Solving a Line Integral Using Green's Theorem

    I'm having trouble on a line integral. Assuming that the closed curve C is taken in the counterclockwise sense. Use Green's Theorem. \int_C F\bullet dR where F=(x^2 + y^2)i + 3xy^2j and C is the circle x^2 + y^2 = 9 This is what I have done so far... \int_0^{2\Pi} \int_0^3 \-r^2...
  44. P

    Last line integral problem (hopefully)

    Greetings again, Show that for F(x,y)=<2xy-3, x^(2)+4y^(3)+5> the line integral F(x,y).dr is independant of path. Then evaluate the line integral for any curve C with initial point (-1,2) and the terminal point (2,3). Thanks again, you all have been very helpful.
  45. P

    Calculating Work Along a Helix Using Line Integrals

    Greetings All Again, I wanted to thank you for the reply on my other problem, it was indeed very helpful and this is a very strange problem. So here goes : Compute the work done by the force field F(x,y,z) = <4y,2xz,3y> acting on an object as it moves along the helix defined...
  46. W

    Explaining Line Integrals and Gradient in Conservative Fields

    I have a question which asked me to evalute the line integral around the curve x^2+y^2=r^2 (z=z0 (a constant)) of the following vectors: (0, z^2, 2yz) and (yz^2, yx^2, xyz) the first one I get as 0, and the second one I get as: -pi(r*z0)^2 Those answers I'm pretty sure are right...
  47. K

    What the heck does a line integral mean?

    Okay, I've searched PF. I actually found a thread that confirmed some of my assumptions. I've searched the web. But I still want to know what the geometric interpretation of a line integral with respect to x (or y) is. The example that made me want to know was \int y^2 dx + x dy ; It was...
  48. O

    How Do You Calculate Work Done by a Force Field Along Different Paths?

    My problem has force decreasing with F=(1/r^2)r, where F is a vecotr and r is unit vector. i need to find a).work done in moving from a point at r=sqrt(2) to a point at r=2*sqrt(2) by a direct radial path and (b) by a path from (1,1)-->(2,1)-->(2,2). Compare my answers. a)I did direct radial...
  49. G

    Why Do Line and Surface Integrals Yield the Same Result for a Triangle Path?

    Hi All, I have the following electric field E = c(2bxy, x^2+ay^2) where a,b and c are constants 1. I need to find the line integral \oint E \cdot dl where the close integration path is defined by the triangle (0,0) (1,0) (1,1) 2. compute the surface integral \int\nabla \times E \cdot...
  50. O

    Line integral curiosity/confusion

    Heya! I was hoping someone could clear this up for me: how would a line integral be represted graphically? I've always liked calculus because it's easy to visualize (almost all the problems have graphs associated with them) - but I don't quite get how to visualize a line integral. Or is it...
Back
Top