The force function is using Cartesian coordinates.
Find the work done from (1,1) to (4,4) using the following path:
So, what I did was:
Is this right?
It seems to me that if you are heading in the x-direction the work done will only depend on the x component of the force, and if you are heading in the y direction the work done will only depend on the y-component of the force, so you can add where f(y) is the y-component of force and f(x) is the x-component of force. Your answer looks right to me.
I've never seen this method either... But it's the way me and my friends interpreted it given this example:
The force exerted on a body is . The problem is to calculate the work done going from the origin to the point (1,1):
Separating the integrals, we obtain:
The first integral cannot be evaluated until we specify the values of x as y ranges from 0 to 1. Likewise, the second integral requires x as a function of y. Consider the first path:
Then:
since y = 0 along the first segment of the path and x = 1 along the second. If we select the path:
then the integral gives . For this force the work done depends on the choice of path.
That's pretty much all we had to go by. No professor, no class, just us and a book...
We did have this relation:
Using that, the separation of the integrals follows.