Questions regarding generalized work equation

[Obliv]

Hello, I am trying to generalize the work equation and understand the very definition of it. From what I understand, Work is the energy required to displace an object with a force in the direction of the displacement. (also the change in kinetic energy but I'm not going to worry about that yet)
W = \int \vec{F} \cdot {d}\vec{x} This equation makes sense to me. If I were to add bounds to the integral, would they be in 3-D if the displacement and force is in 3-D?
Otherwise
W = \int_{x_i}^{x_f} {F_x} {dx} + \int_{y_i}^{y_f} {F_y} {dy} + \int_{z_i}^{z_f} {F_z} {dz}
This method would get the work with calculus I know of. Is it possible to make bounds in multiple dimensions? If so, how would one solve them? Thank you!

edit: Oh one more question: This equation describes non-conservative work, right? What would describe conservative work? Just this? W = \vec{F} \cdot \vec {d}

 

[axmls]

It describes a force that varies with position, unlike your second formula.

The generalized work integral is a line integral, which you'll be able to solve in the general case after you've taken multvariable calculus. It essentially let's you integrate over whatever path you need to integrate over for the problem.

 

[PhanthomJay]

The generalized equation for
Work = \int F.ds applies to both conservative and non conservative forces. For conservative forces , the work done is equal to - \Delta PE . For work done by conservative forces like springs in particular, performing the work integral where F = -kx yields W = -1/2(kx^2) when the spring stretches from rest to x. This is the negative of the change in its PE. Work is not energy, it causes a change in energy, which might be positive or negative or zero. Net work is change in KE, while non conservative work is change in PE plus change in KE. You can also regard non conservative work as the negative of the change in thermal/other energy.