If In is the approximation of a definite integral 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