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

TL;DR

This paper introduces deleting items and disturbing mesh theorems for Riemann integrals, showing that Riemann sums converge under certain conditions even when parts are removed or the partition is altered, revealing deeper geometric insights.

Contribution

It presents new theorems on Riemann sums that allow for item deletion and mesh disturbance while maintaining convergence, extending the understanding of Riemann integrals.

Findings

Riemann sums converge despite deleting some items

Convergence persists under mesh disturbance with specific conditions

Theorems enhance understanding of geometric intuition of integrals

Abstract

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