A force whose direction or magnitude or both change with time is a variable force .
A constant force is rare .It is the variable force which is more commonly encountered .
Here in figure , a plot of a varying force in one dimension versus displacement `F(x)to (x) ` is shown .
If the displacement `Deltax` is small ,the force F(x) as approximately constant and the work done is then area under area the curve of graph `F(x) to Deltax`.
Total work done is the sum of the areas of shaded rectangles ,which is written as `W = sum_(x_(i))^(x_(f))F(x) Deltax`
Here ,the sum is from intial to final position .
If the displacement are allowed to approach zero . then the number of terms in the sum increases without limit , but the sum approaches a definite value equal to the area under the curve .
` :. ` Woek done in shole path ,
`W = lim_(Deltal to0) sum_(x_(i))^(x_(f))F(x) Deltax`
`= int_(x_(i))^(x_(f)) F(x) dx `
Hence ,where lim stands for the limit of the sum when `Deltax` tends to zero . thus , for a varying force the work done can be expressed as a definite integral of force over displacement .
