Originally Posted by

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

$\displaystyle \displaystyle \int_a^{a+h} f(x) \, dx = f(a)h + \frac{f'(a)}{2} h^2 + \frac{f''(\xi)}{6}h^3 $

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