An object is displaced from positin vector `r_1=(2hati+3hatj)` m to `r_2=(4hati+6hatj)` m under a forc `F=(3x^2hati+2yhatj)N` Find the work done by this force.

Explanation

Work done, `W=int_(r_(1))^(r_(2)) F.dr`
`=int_(r_(1))^(r_(2))(3x^(2)hati+2yhatj).(dxhati+dyhatj+dzhatk)`
`=int_(r_(1))^(r_(2))(3x^(2)dx+2ydy)=[x^(3)+y^(2)]_(((2,3)))^(((4,6)))=83J`
In the above two examples, we saw that while calculating the work done we did not mentionb the path through which the object was displaced.
Only initial and final coordinates were required. It shows that in both the examples, the work done is path independent or work done will be equal to whichever path we follow.
Therefore above two forces in which work done is path independent are example of conservative forces.