Sol. The work done by a force along a curve C is defined by the line integral.

W = …... (i)

Substituting the force , and dividing the integral into two parts, (0, 0) → (2,0) and (2, 0) → (2,4), we obtain

W = + …..(ii)

y is a constant in (0,0) → (2,0) (y = 0) and x is a constant in (2,0) → (2,4) (x = 2). Therefore,

W = …...(iii)

We first write the line in parametric from using formula from analytic geometry:

= = = ts …...(iv)

Where x`,y and z are variables (here z 0) and (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) are two given points on the line (here (0, 0, 0) and (2, 4, 0). Substituting these points in the formula, we obtain:

…...(v)

We substitute in , so that = 12t 2 + 24t 2 . Therefore,

W = …...(vi)

(we used dx = 2dt and dy = 4dt).

We will use parameterization:

…...(vii)

So that the force is (t) = (t 4 – t 2 ) – 3t 3 . Therefore,

W = …...(viii)

Conclusion – The force is not conservative, as the work is dependent on the path taken.