Questions about Line Integrals

In summary: Parametrization allows for a more general representation of the path and accounts for any changes in the direction of the path. It is also necessary for certain types of integrals, such as line integrals. In summary, parametrization is necessary for accurately calculating the work done on a particle traversing a path, as it allows for a more comprehensive representation of the path. Simply plugging in the changes in x and y for the limits of integration may not accurately account for changes in direction or other factors.
  • #1
Noone1982
83
0
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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  • #2
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?
 
  • #3
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?
 
  • #4
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.
 

1. What is a line integral?

A line integral is a type of integral used in multivariable calculus to calculate the total change of a scalar or vector field along a given path. It involves breaking down the path into small segments and calculating the contribution of each segment to the overall integral.

2. What is the difference between a line integral and a regular integral?

A regular integral calculates the area under a curve on a two-dimensional plane, while a line integral calculates the total change along a path in a three-dimensional space. Line integrals also take into account the direction of the path, whereas regular integrals do not.

3. How is a line integral evaluated?

A line integral is evaluated by breaking down the path into small segments, calculating the contribution of each segment, and then adding them together to get the total change. This can be done using various methods such as the Fundamental Theorem of Calculus, Green's Theorem, or Stoke's Theorem.

4. What are some real-life applications of line integrals?

Line integrals have various practical applications in fields such as physics, engineering, and economics. They can be used to calculate work done by a force along a specific path, electric field strength in a given region, and even the cost of a given journey with varying prices along the way.

5. Can line integrals be applied to curved paths?

Yes, line integrals can be applied to paths that are not straight lines. In fact, they are often used to calculate changes along curved paths in real-life scenarios. However, the calculations may become more complex as the path becomes more curved, and certain techniques such as parametrization may be required.

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