Differential length example - should it be negative in a_x or a_y directions?

The answer to this issue is you need to be consistent with your notation. As given by Cheng in his textbook, all differential lengths are defined as positive. For example in Cartesian coordinate, we have

 
 Let us take an example as given by the attachment below:

 

The solution on the LHP is not correct as the direction is already specified in the negative x-direction which should already be taken care of by the integration limit. It is as if the integration limit (from 2 to 1) and the direction of your for the first path should result in positive direction which is not consistent with the resulting direction. Unless, you integrate from 1 to 2, the result will be correct.

 As I hightlighted in class, this is different from the differential surface area in which the direction is outward normal to the surface area. Hence, you have either positive or negative depending on the box surface (for Cartesian example).