no its not the average over ∂t, its the mean of two numbers, u and v, initial velocity and current velocity (this applies only to a constant acceleration).

If you look at http://en.wikipedia.org/wiki/Equations_o...equation_2

it stating the equation v bar = s/t (thought it calls v bar average velocity), which for a constant acceleration alone = (u+v)/2.
This was all known before calculus techniques and does not make any assumptions about graphs or areas under graphs.

as I showed in a previous post the two equations
v = u + at
s = 1/2 * (u + v) * t

are both derived from basic mathematics and the resulting formula for t in terms of s and a was derived from simple algebraic manipulation.

so you dont need calculus for somthing this simple!