with step 
using the trapezoidal rule (respectively Simpson's rule),
if Ca,b denotes the number of significant digits common to a and b, we show,
in this paper, that 
According to the previous theorems, using the CADNA library which
allows on computers to estimate the round-off error effect on any
computed result, we can compute dynamically the optimal value of n to
approximate I and we are sure that the exact significant digits of In
are in common with the significant digits of I.
Keywords:
numerical validation, quadrature methods,
trapezoidal rule, Simpson's rule