     Calculus Animations,Graphics and Lecture Notes Line Integrals    Home About Kelly Liakos Sponsors Calculus 1- Limits and Derivatives Calculus 1 - The Second Derivative Calculus 1 and 3 Formula Sheets Chain Rule Computer Lab Assignments Conic Sections Differential Equations Directional Derivatives/Gradient Double Integrals Equillibrium Solutions 1st order DEs 1st order Diff Eqs -Motion Flux Integrals and Surface Integrals Infinite Sequences Infinie Series Level Curves and Level Surfaces Line Integrals Optimization and Related Rates Optimization for Functions of 2 Variables Parametric Equations 2-space Parametric Equations 3-space Partial Derivatives Polar Coordinate System Polar Coordinates- Derivatives and Integrals PreCalculus Riemann Sums and the Fundamental Theorem of Calculus 2d order Diff EQS-Motion 2d Partial Derivatives Supplemental Exercises and Solutions Tangent Planes/ Differential for f(x,y) Trigonometry Trigonometry Applications Triple Integrals Unit Tangent Vectors/Unit Normal Vectors Vectors in General Vector Valued Functions Visualizing Limits of Functions of 2 Variables Work Links Special Topics Scratch Paper 2 Poems In Physics1 or Calculus 2 you were introduced to the idea of Work. (See the page on Work if you want a review)

If an object is moving in one-dimension under the influence of a force which is constant in both magnitude and direction then work is defined by Work = Force x Distance.

In Calculus2 you examined the situation when F was not constant in magnitude but was constant in direction. Motion was still in one dimension. This led to an integral formula for work.

However what happens if we have a vector field  F neither constant in magnitude nor direction and a curve in 2 or 3-space defined parametrically by r(t). How do we calculate the work done in this more general setting?

This leads us to the development of the Line Integral also known as the Contour Integral.

In the first download we develop the Line Integral and the following 6 animations illustrate the general situation.

We then present the solutions to the line integrals in the 6 animations followed by further examples.

Before beginning you may want to review the notes on vector fields.

Notes - Vector Fields Notes - Development of the Line Integral

In the first 3 animations we use the vector field F = -y i + x j.

In the first animation the path is the unit square.

In the second animation the path starts at the origin, The particle moves along the parabola

y = x2 to the point (1,1) then moves back to the origin along  y = x.

In the third animation the path is a triangle with vertices at (0,0), (1,0), and (1/2,1/2).

In the animations 4 through 6 we change the vector field to F = y i + x j  but use the same paths as in the first 3.

Animation 1 - Unit Square

Animation 2 - Parabolic Path

Animation 3 - Triangular Path

Animation 4 - Unit Square

Animation 5 - Parabolic Path

Animation 6 - Triangular Path

Solutions to Animations 1-3

Solutions to Animations 4-6

In the previous examples we proceeded as if the parameterization of a particular curve was unimportant. The following download demonstrates that indeed this is the case.

In this download we also discuss what happens if we reverse the orientation along a given path.

Notes - Independence of Parameterization

Path Independence and Conservative Vector Fields

Green's Theorem

 Show that if F is a force field with constant magnitude k pointing outward from the origin  then the work done as a particle travels along the smooth curve y = f(x) as x varies from a to b  is k ((b2 +f(b)2)1/2 - (a2 +f(a)2)1/2)

A 3-D example using direct computation and Stoke's Thm

Another Example in 3-D with direct computation and Stoke's Theorem

For more examples involving Stokes Theorem  see the page on Flux Integrals   The Smart Bunny-A very short story by Kurt Vonnegut Jr.  