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1·It is difficult for learner to understand the concept of Riemann integral which is defined by using the limit of Riemann sum.

用积分和的极限定义的黎曼积分对于初学者来说是一个很难理解的概念。

2·In this paper we give some simpler definitions of definite integral, and show that they are all equivalent to the definition of Riemann integral.

本文给出了定积分的几个较简单的定义，并证明这些定义均与黎曼积分定义等价。

3·Based on Darboux theory, this paper discussed the integrability of the Riemann Integral and provides a necessary and sufficient condition for integrability.

文章利用达布和理论，讨论了黎曼积分的可积性问题，给出了一个可积的充分必要条件。

4·This paper sums up the common character of the concept of many integrals, abstracts the concept of Riemann integral and gives the integral conditions of the Riemann integral.

分析了诸多积分概念的共性，抽象出黎曼积分的定义，给出了黎曼可积的条件。

5·After constrcting the perfective space prove that this space is just the space of lebes gue integratiable function thus explain that lebes gue integral is the form of the perfective riemann integral.

在构造了完备化空间之后，证明了该空间就是勒贝格可积函数空间，从而说明了黎曼积分的完备化形式是勒贝格积分。