The example of calculating the work performed by the drag force when raindrops fall pre-empted a careful analysis of what to do when the force varies as the particle moves. Nonetheless it indicates to us that we must be able to come up with a suitable definition of work for this case as well.
The trick is to divide the motion into intervals so small that the force
for all intents and purposes is constant over each mini-interval
so that we are allowed to apply the simple form of the work kinetic
energy theorem in this interval of the motion
Notice the important order in the subtraction of kinetic energies: The work put
in is the kinetic energy gained.
Summing up all these equations we get
We have re-found the work-kinetic energy theorem and therefore
happily accept the corresponding definition of work performed
by a spatially varying force:
In the last equality I have used the mathematicians definition of an integral
Consult your math-book or come to my office
hours if you have forgotten or did never learn about integration.