Could somebody please explain how, from the definition of integral, we can deduce the following:

Obviously some form of the Taylor expansion with Lagrangian Remainders is going on, but where does the multiplying by h on the RHS come into it, why are the numerators moved 'back' one term than they usually are in a Taylor expansion, and how do we arrive at this result 'from the definition of integral'?

AnonymitySquared