Abstract
In this paper, through a direct computation with subintervals partitioning [0, 1], we compute better a posteriori bounds for the average case error of the difference between the true value of <TEX>$I(f)=\int_{0}^{1}f(x)dx$</TEX> with <TEX>$f{\in}C^r$</TEX>[0, 1] minus the composite trapezoidal rule and the composite trapezoidal rule minus the basic trapezoidal rule for <TEX>$r{\geq}3$</TEX> by using zero mean-Gaussian.
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