I understand how the problem is solved, the only issue I have is I don't understand how they are getting the units. As in how did they figure out it was kip*ft^3?
Simple - The book's got the units wrong ...
... if it's as simple a "units" problem as it looks.
It happens, typos in manuscripts - or the author had a brain-fart that day; just 'cos it's in print and you might have paid a lot for it (it looks like an expensive book) doesn't mean everything in it is right.
If I understand it, the question's asking for the shape of the curve of the beam. If the "independent" variable x is in feet, then so is v - the deflection along any chosen point (presumably) on the curved beam (well, thinking about it a bit, probably only any chosen point to the left of the ?roller? bearing).
Do you happen to know, as you seem to be in this field, if any preferably simple combination of moments/forces applied to a rod will constrain its shape accurately to that of a parabola? I was thinking of working it out "backwards" - working out the (sum of) moment(s) that would be required to produce y = x², and in addition trying to use the exact formula for the curvature, ie: dθ/ds where dθ is the angle subtended by element of arc-length ds. Then trying to figure out if any possible (again preferably simple) arrangements of real physical moment(s) can be arranged so as to be equal to the (sum of) moment(s) calculated.