Final Answer:
The work done by the force F as the object moves from the origin to the point (a, b) is given by the line integral of F along the path taken by the object.
Step-by-step explanation:
The work done (W) by the force F along a path can be calculated using the line integral formula:
W = ∫(2xy + 3yx) dx + ∫(2xy + 3yx) dy
where F is the force vector, dr is the differential displacement vector along the path, and the integration is performed over the path C.
In this scenario, the force is given by F = (2xy + 3yx) i + (2xy + 3yx) j, and the path is described as moving first along the x-axis to (a, 0), then parallel to the y-axis to (a, b). Let's denote the position vector along the path as r = (x, y).
The work done along the x-axis segment is calculated as:
W_x = ∫(2xy) dx from 0 to a
The work done along the y-axis segment is calculated as:
W_y = ∫(3yx) dy from 0 to b
Finally, the total work done is the sum of W_x and W_y:
W = W_x + W_y
The specific calculations involve integrating the given expressions with respect to x and y over their respective intervals.