How Do I Calculate Work Using Line Integrals with Multiple Paths?

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Discussion Overview

The discussion revolves around calculating work using line integrals with multiple paths in a vector field. Participants explore the application of parametric equations and the integration limits for different segments of the path.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a vector field and describes a path for calculating work, asking how to set limits for the integral.
  • Another participant suggests using parametric equations and indicates that the integral should be expressed in terms of a single parameter, recommending two separate integrals due to a change in the path at x=5.
  • A participant questions the necessity of parameterization, proposing that it might be possible to directly use changes in x and y for limits of integration for each curve.
  • Another participant reiterates the previous point about the possibility of not needing parameterization but notes that this approach may not be generally applicable.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of parameterization for calculating work along the given paths, indicating that there is no consensus on the best approach.

Contextual Notes

There is uncertainty regarding the applicability of direct integration limits versus parameterization, and the discussion does not resolve the mathematical steps involved in the integration process.

Noone1982
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Say we have a vector, let's use something simple like

A = 2xyi + 3yzj

Say we want to find the work done on a particle traversing a path, so we just add up the work done on each path. Let the path be:

y = x^2 from x = 0 to x = 5
y = 25 from x = 5 to x = 10
now a final path from (10,25) to (10,35)

How do I enter my limits in if Work = integral A • Ds ?
 
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Write the path in parametric equations. Use the value of the parameter corresponding to the beginning and ending points.
In this particular case, since y is a function of x, it is simplest to take x itself as the parameter. Write your integral entirely in terms of x and use the x values as limits of integration. Since your function changes at x= 5, you will probably want two integrals, one from 0 to 5, the other from 5 to 10.

By the way, your force vector is A = 2xyi + 3yzj which includes "z" but you only have i and j and your path only includes x and y. Was that intentional?
 
No, I was just making up a random vector.

Must I do a parametization? Is it not possible just to plug in the change of x for the limits of dx and the change of y for the limits of dy for each separate curve then add 'em up?
 
Noone1982 said:
No, I was just making up a random vector.

Must I do a parametization? Is it not possible just to plug in the change of x for the limits of dx and the change of y for the limits of dy for each separate curve then add 'em up?
This may sometimes be possible, but in general it's not.
 

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