Your question is very clear : you want to compute the second level integral of the acceleration, which theorically is possible, by supposing the speed null at some time, but I really doubt you could get something precise enough to make any sense (as in many integral computations).
This isn't done today because the error is too big. Done in hardware (for permanent integration of the acceleration) it could be a little more precise but probably not enough to really compute a distance in any acceptable sense of the word "accuracy".
If you want to try it by yourself, here's a document describing more in detail the approach : http://perso-etis.ensea.fr/~pierandr/cours/M1_SIC/AN3397.pdf