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

[Noone1982]

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 ?

 

[HallsofIvy]

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?

 

[Noone1982]

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?

 

[TD]

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.