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Based on the definition of Riemann definite integral,deleting items anddisturbing mesh theorems on Riemann sums are given. After deleting some itemsor disturbing the mesh of partition, the limit of Riemann sums still convergesto Riemann definite integral under specific conditions. These theorems can dealwith a class of complicate limitation of sum and product of series of afunction, and demonstrate that the geometric intuition of Riemann definiteintegral is more profound than ordinary thinking of area of curved trapezoid.

Deleting Items and Disturbing Mesh Theorems for Riemann Definite Integral and Their Applications