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. E

    Line Integral Notation wrt Scalar Value function

    I'm getting a bit confused by the specific notation in the question and am unsure what exactly it is asking here/how to proceed. Homework Statement Given a scalar function ##f## find (a) ##∫f \vec {dl}## and (b) ##∫fdl## along a straight line from ##(0, 0, 0)## to ##(1, 1, 0)##.Homework...
  2. R

    Line integral of vector field from Apostol calculus

    Homework Statement Here are the three problems that i couldn't solve from the book Calculus volume 2 by apostol 10.9 Exercise 2. Find the amount of work done by the force f(x,y)=(x^2-y^2)i+2xyj in moving a particle (in a counter clockwise direction) once around the square bounded by the...
  3. P

    MHB Luca's question via email about a line integral....

    I am assuming that this line integral is along the straight line from $\displaystyle \begin{align*} (0,0,0) \end{align*}$ to $\displaystyle \begin{align*} \left( 5, \frac{1}{2}, \frac{\pi}{2} \right) \end{align*}$, which has equation $\displaystyle \begin{align*} \left( x, y, z \right) = t\left(...
  4. R

    Line integral problems in Apostol calculus

    Homework Statement A two dimensional force field f is give by the equation f(x,y)=cxyi+x^6 y^2j, where c is a positive constant. This force acts on a particle which must move from (0,0) to the line x=1 along a curve of the form y=ax^b where a>0 and b>0 Homework Equations Find a value of a(in...
  5. Toby_phys

    Line Integrals around a Square on the x-y Plane

    Homework Statement Evaluate the following line integrals, showing your working. The path of integration in each case is anticlockwise around the four sides of the square OABC in the x−y plane whose edges are aligned with the coordinate axes. The length of each side of the square is a and one...
  6. lep11

    Calculate the following line integral

    Homework Statement Let ##f(x,y)=(xy,y)## and ##\gamma:[0,2\pi]\rightarrowℝ^2##,##\gamma(t)=(r\cos(t),r\sin(t))##, ##r>0##. Calculate ##\int_\gamma{f{\cdot}d\gamma}##. Homework EquationsThe Attempt at a Solution The answer is 0. Here's my work. However, this method requires that you are...
  7. F

    Correcting Errors in Conservative Line Integral Calculation

    Homework Statement I am having question with part c , for both c1 and c2 , here's my working for c1 , i didnt get the ans though . My ans is -5 , but the given ans for c1 and c2 is 27 , is the ans wrong ? Or which part i did wrongly ? Homework EquationsThe Attempt at a Solution
  8. S

    I Vector Calculus: What do these terms mean?

    In our section on path independence, we were asked to find the potential function given a vector field. Our teacher says to use only line integrals to find the potential function, and not any other method. Like if we have ##F=\left< M,N,P \right> ## The first step is to determine if the domain...
  9. garylau

    Where is pi/4 coming from in the line integral?

    Sorry where is pi/4 coming from in the line integral(section 3)? because i think it should be 1/2=tan(theta) which theta is 26.5651... it is impossible that the angle is pi/4? where is pi/4 coming from inside the circle? thank
  10. S

    Basic Line-Integral: Just trying to know what is being asked

    Hello. I'm new to physics, and the problem I have seems so basic, mathematically speaking. I'm just failing to grasp exactly what is being asked. If I can find that, I believe I can find the answer. Here it is: 1. Homework Statement Let A = x2ˆx + y2ˆy + z2ˆz Consider the parabolic path y2 =...
  11. O

    I Solving Line Integral Limits: Negative Result?

    I want to the line integral in the following picture: The field is the blue arrows that go left to right, and the path is the orange line that is going from right to left. Just by looking at the picture, it is clear that the result will be negative, but when I set up the integration this is...
  12. S

    Line integral across a field given by circular distribution

    Homework Statement Evaluate \int_C \vec{F} \cdot d\vec{r} Where \vec{F} is the field generated from a circular thread of radius b in the xy plane, with magnitude j in the direction \hat{\varphi} (i.e. not along the curve, I take it) C: (x,y,z) = b(1+ \cos{\alpha}, 0, \sin{\alpha}) 3. The...
  13. C

    Line integral of scalar field ( piecewise curve)

    Homework Statement for the line segment c2 , why did the author want to differentiate dx with respect to dy ? and he gt dx = 0 ? I'm curious why did he didnt do so for C3 , where dy= 0 ..Why didnt he also differentiate dy with dx ? dy/dx = 0 ? Homework EquationsThe Attempt at a Solution is...
  14. C

    Line Integral Homework: How to Solve for the Total Length of a Curve

    Homework Statement i'm not sure what is line integral... Homework EquationsThe Attempt at a Solution Does it mean total length of line under the curve?
  15. S

    Line integral convert to polar coordinates

    Homework Statement I need to find the work done by the force field: $$\vec{F}=(5x-8y\sqrt{x^2+y^2})\vec{i}+(4x+10y\sqrt{x^2+y^2})\vec{j}+z\vec{k}$$ moving a particle from a to b along a path given by: $$\vec{r}=\frac{1}{2}\cos(t)\vec{i}+\frac{1}{2}\sin(t)\vec{j}+4\arctan(t)\vec{k}$$ The Attempt...
  16. T

    Line integral over vector field of a shifted ellipse

    This is part of a larger question, but this is the part I am having difficulty with. I have had an attempt, but am not sure where I am making a mistake. Any help would be very, very appreciated. 1. Homework Statement Let C2 be the part of an ellipse with centre at (4,0), horizontal semi-axis...
  17. S

    Line Integral over circle region

    Homework Statement Evaluate ∫c (x + y) ds, where C is the circle centred at (1/2, 0) with radius 1/2. Homework EquationsThe Attempt at a Solution parametrise x=1/2cos(t) y=1/2sin(t) 0≤t≤2π ds=√dx2+dy2 =√(1/2)2-sin2(t)+(1/2)2cos2(t) =√-(1)2(1/2)2sin2(t)+(1/2)2cos2(t)...
  18. S

    Evaluate length of the spiral (Line Integral)

    Homework Statement Evaluate the length of the spiral with parametric equation ψ(t) =< 2 cost, 2 sin t, π/t >, with t ∈ [0, 2π]. Homework Equations Line integral ∫C f(x,y) dS The Attempt at a Solution f(x,y) = z = π/t ∫C π/t dS [0, 2π] are the lower and upper bounds of integration dS=...
  19. JulienB

    Calculating Work with Line Integrals: Solving a Force Field Problem

    Homework Statement Hi everybody! I would like to make sure I properly solved that problem because I find the result strange: Given a force field ##F_x = axy^3##, ##F_y = bx^2y^2##, ##F_z = cz^3##. Calculate the work with the line integral ##\int_{C} \vec{F} \cdot d\vec{r}## from point...
  20. S

    How Do You Calculate Work Done in a Vector Field Along a Parametric Path?

    Fine the word done in moving a particle in the force field F=<2sin(x)cos(x), 0, 2z> along the path r=<t,t,t2>, 0≤t≤π To do the line integral, I need to find F(r(t)), but I don't understand how to express it. For example I looked at the online notes provided here...
  21. S

    Line integral over a vector field

    Homework Statement Evaluate ∫C < −y, x − 1 > dr where C is the closed piecewise continuous curve formed by the line segment joining the point A(− √ 2, √ 2) to the point B( √ 2, − √ 2) followed by the arch of the circle of radius 2, centered at the origin, from B to A. 2. The attempt at a...
  22. Destroxia

    Calculating Surface Area Using a Line Integral: A Case Study

    Homework Statement Use a line integral to find the area of the surface that extends upward from the semicircle ##y=\sqrt{9-x^2}## in the ##xy##-plane to the surface ##z=3x^4y## Homework Equations Parametric Equation for Circle: ## x = rcos(t) ## ## y = rsin(t) ## Line Integral: ## \int_c...
  23. S

    Line Integrals: What Do They Represent and How Can They Be Visualized?

    Homework Statement Need to visualize what it means by Line Integral along curve C with respect to x or y axis. For example suppose the curve is C (I did not find a way to write the C under the integration sign here) ∫ f(x,y) ds is like a fence along C whose height varies as per f(x,y). The...
  24. M

    Calculation of Line Integral over a Parabola and Straight Line

    Homework Statement Evaluate line integral(x+sqrt(y)) over y=x^2 from (0,0) to (1,1) and y=x from (1,1) to (0,0) Homework Equations n/a The Attempt at a Solution I set up integral from 0 to 1 of 2t(sqrt(1+4t^2))dt for the parabola part and then added integral from 0 to 1 of...
  25. E

    Line Integral Problems: Solving for Work and Potential Functions

    I'm used to parameterizing however I'm not sure how to solve these types of problems, any help would be much appreciated. 1) Calculate the line integral ∫v⋅dr along the curve y=x3 in the xy-plane when -1≤x≤2 and v=xyi+x2j 2) a) Find the work that the force F = (y2+5)i+(2xy-8)j carries...
  26. qq545282501

    What is the Line Integral of xydx+4ydy along a Curve from (1,2) to (3,5)?

    Homework Statement \int xydx+ 4ydy where C is the curve from (1,2) to (3,5) made up of the twoline segments parallel to the coordinate axes. c_1:(1,2)\rightarrow(3,2) c_2:(3,2)\rightarrow(3,5) Homework EquationsThe Attempt at a Solution i got c2 correct, y=2+3t, and x = 0, for t goes from 0...
  27. T

    How is a line integral over any closed surface 0?

    We just started going over line integrals in calculus, and have been told that the integral over any closed surface is 0. What I don't get is then why can we do the line integral of a circle to get 2##\pi##r? Since a circle is a closed surface, shouldn't the line integral then be 0?
  28. I

    Line integral in spherical coordinates

    Homework Statement The vector field ##\vec B## is given in spherical coordinates ##\vec B(r,\theta,\phi ) = \frac{B_0a}{r\sin \theta}\left( \sin \theta \hat r + \cos \theta \hat \theta + \hat \phi \right)##. Determine the line integral integral of ##\vec B## along the curve ##C## with the...
  29. A

    Stokes theorom question with a line

    Homework Statement F[/B]=(y + yz- z, 5x+zx, 2y+xy ) use stokes on the line C that intersects: x^2 + y^2 + z^2 = 1 and y=1-x C is in the direction so that the positive direction in the point (1,0,0) is given by a vector (0,0,1) 2. The attempt at a solution I was thinking that I could decide...
  30. SquidgyGuff

    Calcularing area vector using line integral

    Homework Statement A closed curve C is described by the following equations in a Cartesian coordinate system: where the parameter t runs monotonically from 0 to 2π, thus defining the direction of C. Calculate the area vector of the planar region enclosed by C, using the formula: 2. The...
  31. fricke

    Calculating Line Integrals on the Surface of a Sphere

    What's the line integral of sphere? Is it possible to get the line integral in three dimensions? What kind of line are we integrating?
  32. kostoglotov

    Insight into determinants and certain line integrals

    I just did this following exercise in my text If C is the line segment connecting the point (x_1,y_1) to (x_2,y_2), show that \int_C xdy - ydx = x_1y_2 - x_2y_1 I did, and I also noticed that if we put those points into a matrix with the first column (x_1,y_1) and the second column (x_2,y_2)...
  33. kostoglotov

    Line Integral Example - mistake or am I missing something?

    This is an example at the beginning of the section on the Fundamental Theorem for Line Integrals. 1. Homework Statement Find the work done by the gravitational field \vec{F}(\vec{x}) = -\frac{mMG}{|\vec{x}|^3}\vec{x} in moving a particle from the point (3,4,12) to (2,2,0) along a piece wise...
  34. nuuskur

    Line integral of the second kind

    Homework Statement Given the polar curve r = 3\sqrt{\cos{2\varphi}},\ -\frac{\pi}{4}\leq\varphi\leq\frac{\pi}{4}. Find the area of the surface enclosed by the curve using line integral of the second kind. Homework EquationsThe Attempt at a Solution According to Green's theorem: if F(x,y) and...
  35. S

    Work Done line Integral question - Electrostatics - help please

    Can anyone please tell me where I am going wrong? I am getting the incorrect answer for the Word Done should be: WD = q(3x^2-6y) ... Apologies for not changing it into the format on here - but for my revision I have pretty much done that myself.
  36. Calpalned

    How Does Green's Theorem Simplify Calculating a Line Integral for an Ellipse?

    Homework Statement Use Green's Theorem to evaluate the line integral along the given positively oriented curve. ##\int_C y^4 dx + 2xy^3 dy ##, C is the ellipse ##x^2 + 2y^2 = 2##. Homework Equations Change of variables: ##\int \int_S f(x(u,v),y(u,v)) |{\frac {\partial(x,y)}{\partial (u,v)}}|...
  37. Calpalned

    Using Green's Theorem for line integral

    Homework Statement Use Green's Theorem to evaluate the line integral along the given positively oriented curve. 1) Is the statement above the same as finding the area enclosed? 2) ##\int_C \cos ydx + x^2\sin ydy ##, C is the rectangle with vertices (0,0) (5,0) (5,2) and (0,2). 3) ##\int_C y^4...
  38. nmsurobert

    Line integral of a vector field

    Homework Statement Consider the vector field F(r) = Φ^ (a) Calculate ∫ F⋅dl where C is a circle of radius R (oriented counterclockwise) in the xy-plane centered on the origin. Homework Equations maybe Φ^ = -sinΦx^ + cosΦy^ The Attempt at a Solution not really a solution. i am just stuck at...
  39. ognik

    Magnetic moment of current loop integral

    The question states: The calculation of the magnetic moment of a current loop leads to the line integral ∮ r x dr I am puzzled - shouldn't this be ∮ r x dl where r is the radius of the loop and dl is the small change along the loop? (I think dr would be in the same direction as r, so no cross...
  40. N

    How Do You Evaluate Line Integrals for Different Paths?

    Homework Statement Evaluate the line integral ∫^{(1,0)}_{(0,1)} (x^2-y)dx + (y^2+x)dy along (a) a straight line from (0,1) to (1,2); (b) the parabola x=t, y=t2 + 1; (c) straight lines from (0,1) to (1,1) and then from (1,1) to (1,2). Homework Equations Equation of line: y = mx + c The Attempt...
  41. P

    Using Green's Theorem for Solving Line Integrals

    Homework Statement i have this problem and need your help. I tried to solve the first 2 question but don't know ho to solve the third one
  42. A

    Complex Contour Integral Problem, meaning

    Homework Statement First, let's take a look at the complex line integral. What is the geometry of the complex line integral? If we look at the real line integral GIF: [2]: http://en.wikipedia.org/wiki/File:Line_integral_of_scalar_field.gif The real line integral is a path, but then you...
  43. Amy Marie

    Evaluating Line Integral with Green's Theorum

    Homework Statement Use Green's Theorum to evaluate the line integral ∫c (x^2)y dx, where c is the unit circle centered at the origin. Homework EquationsThe Attempt at a Solution Taking the partial derivative with respect to y and subtracting it from zero(I'm taking the dy in the original...
  44. dwn

    Line Integral of f(x,y,z): Exploring the Answer

    Homework Statement Find the line integral of f(x,y,z) = x+y+z over the straight-line segment from (1,2,3) to (0,-1,1). Homework Equations ∫ f(x,y,z)ds = ∫ f(g(t), h(t), k(t)) |v(t)| dt The Attempt at a Solution I arrived at the correct solution, but I'd like some clarity on the result...
  45. N

    Line integral over a given curve C

    Homework Statement Evaluate the line integral over the curve http://webwork.math.ttu.edu/webwork2_files/tmp/equations/33/92ca0e3b907f876e5a974ad1457d1f1.png from to http://webwork.math.ttu.edu/webwork2_files/tmp/equations/6f/bccf9dd59b9c22450c590a042bb77d1.png . ∫-ydx+3xdy (over the curve C)...
  46. blair chiasson

    I have a problem with variable forces

    Homework Statement A force F has components F sub x = axy-by^2, F sub y= -axy+bx^2 where a= 2N/m^2 & b=4N/m^2 Calculate the work done on an object of mass 4kg if it is moved on a closed path from (x,y) values of (0,1) to (4,1), to (4,3) to (0,3) and back to (0,1). all coordinates are in metres...
  47. AwesomeTrains

    Line Integral - Stokes theorem

    Homework Statement Hello I was given the vector field: \vec A (\vec r) =(−y(x^2+y^2),x(x^2+y^2),xyz) and had to calculate the line integral \oint \vec A \cdot d \vec r over a circle centered at the origin in the xy-plane, with radius R , by using the theorem of Stokes. Another thing, when...
  48. G

    Solving a Line Integral Problem: Struggling for the Right Answer

    Hi everyone. I ran into a minor problem while trying to solve a problem on line integral. I suspect this question to be very straight-forward as it gave the parametric equations of the curve C. However, I am still unable to get the answer for some reason. May I have someone to point out where I...
  49. I

    Line integral around a circle, using polar coordinates

    Given the force (derived from a potential in planar polar coordinates) F(p,w) = 2p+sin(w)e_p+cos(w)e_w Where e_p and e_w are unit vectors How do I calculate the line integral over a circumference that is defined as: p = 2 0 ≤ w ≤ 2pi Using the definition of a line integral \int_0^{2pi} \...
  50. L

    Line integral over vector field

    Homework Statement Let F be a vector field F = [-y3, x3+e-y,0] The path in space is x2 + y2 = 25, z = 2. My parametrization is r(t) = [5cos(t),5sin(t),2] Homework Equations Line integral is the integral of F(r(t)) * r'(t) dt, where here the asterisk * is for the dot product, not normal...
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