# What is Line integral: Definition and 401 Discussions

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. ### Line integral of a vector field (Polar coordinate)

Hi, I am not sure if I have solved task b correctly According to the task, ##\textbf{F}=f \vec{e}_{\rho}## which in Cartesian coordinates is ##\textbf{F}=f \vec{e}_{\rho}= \left(\begin{array}{c} \cos(\phi) \\ \sin(\phi) \end{array}\right)## since ##f \in \mathbb{R}_{\neq 0}## is constant...
2. ### Line Integral Solution for Curve γ: Simplifying Substitutions

Hello, How should I go about to solve this line integral along the line curve γ? I attempt to apply this relation but the substitutions get too messy. Thanks

37. ### B Line Integral, Dot Product Confusion

From my interpretation of this problem (image attached), the force applied to the point charge is equal and opposite to the repulsive Coulomb force that that point charge is experiencing due to the presence of the other point charge so that the point charge may be moved at a constant velocity. I...
38. ### I Force fields in curvilinear coordinate systems

I am trying to solve problems where I calculate work do to force along paths in cylindrical and spherical coordinates. I can do almost by rote the problems in Cartesian: given a force ##\vec{F} = f(x,y,z)\hat{x} + g(x,y,z)\hat{y}+ h(x,y,z)\hat{z}## I can parametricize my some curve ##\gamma...
39. ### Line Integral for Electromagnetic Force

http://web.mit.edu/sahughes/www/8.022/lec01.pdf So I'm trying to understand how to get from F = ∫[(Q*λ)*dL*r]/(r^2) to F=∫q*λ*[(xx+ay)/(a^2+x^2)^(3/2)]*dx Like I don't understand why the x and y components of r are negative, or why "The horizontal r component is obviously zero: for every...
40. ### I Area between two closed curves

I have been trying to find an answer to this problem for some time. So I was hoping the community might be able to point me in the right direct. https://drive.google.com/open?id=1Y2hYkRG94whLroK20Zmce07jslyRL3zZ I couldn't get the image to load. So above is a link to an image of the problem...
41. ### Calculating Line Integral in xy-Plane

Homework Statement Calculate the line integral ° v ⋅ dr along the curve y = x3 in the xy-plane when -1 ≤ x ≤ 2 and v = xy i + x2 j. Note: Sorry the integral sign doesn't seem to work it just makes a weird dot, looks like a degree sign, ∫.2. The attempt at a solution I have to write something...
42. ### MHB 16.1.9 Line Integral over space curves

Evaluate $\displaystyle \int_C(x+y)ds$ where C is the straight-line segment $x=t, y=(1-t), z=0,$ from (0,1,0) to (1,0,0) ok this is due tuesday but i missed the lecture on it so kinda clueless. i am sure it is a easy one.
43. ### MHB Multivariable calculus line integral work

calculate the work done by the force field $F(x,y)=(ye^{xy})i+(1+xe^{xy})j$ by moving a particle along the curve C described by gamma (γ):[0,1] in $R^2$, where gamma (γ)=(2t-1, t²-t)
44. ### Line integral of a curve

Homework Statement Homework EquationsThe Attempt at a Solution Line integral of a curve ## I = \int_{ }^{ } yz dx + \int_{ }^{ } zx dy + \int_{ }^{ } xy dz ## with proper limits. ## I = \int_{\frac { \pi }4}^{ \frac { 3 \pi} 4} abc ( \cos^2 t - \sin^2 t ) dt = -abc ## |I| = abc...
45. ### Line integral of a vector field

Homework Statement [/B] I would like to ask for Q5b function G & H Homework Equations answer: G: -2pi H: 0 by drawing the vector field The Attempt at a Solution the solution is like: by drawing the vector field, vector field of function G is always tangential to the circle in clockwise...
46. M

### Using Green's Theorem for a quadrilateral

Homework Statement Evaluate the line integral of (sin x + y) dx + (3x + y) dy on the path connecting A(0, 0) to B(2, 2) to C(2, 4) to D(0, 6). A sketch will be useful. Homework Equations Sketching the points, I have created a parallelogram shape. I also know that green's theorem formula, given...
47. ### I Evaluation of a line integral

calculate the line integral for a vector field F= -xy⋅j over a circle which is c(t)=costi+sintj, so I used x=cost y=sint and ∫(0 to 2pi) -(sintcost)(cost)dt=(cos^3(2pi)-cos^3(o))/3=0 now here is the problem, if this enclosed line integral is zero then why is the vector field not conservative?
48. ### 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...
49. ### 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...
50. ### 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(...