Difficulty following example in Taylor's Classical Mechanics

[cwbullivant]

http://i.imgur.com/GP6QorG.jpg

I don't follow the integration in it.

I'm assuming F_x(x,0) and F_y(1,y) are the partial derivatives of F with respect to x and y, respectively, but given that, I can't seem to get my head around the result where the partial with respect to x is F_x = (x, 0) instead of F_x = (0, 2), and similar for y.

 

[Delta2]

They are not the partial derivatives , they are the x and y component of the F force. In the straight path from O to Q its only the x component that does work (y component is always perpendicular). In the path from Q to P its only the y component that does work.

The x component in the path from O to Q is zero because the force is F=(y,2x) so the x component is simply y but y=0 along this path. Similarly the y component from Q to P is simply 2x, but x=1 always for this path so the y component is 2.