关于积分第一中值定理的逆命题

On the Inverse Proposition of the First Mean Value for Integrals

著名的积分第一中值定理在《数学分析》中占有十分重要的位置,作为很多学科计算的一个重要工具,它得到了多种形式的改进和推广。但积分中值定理的逆命题一般不成立,经较深入地讨论它的逆命题,通过加强条件,给出成立的情形,得出相关定理并给予证明。在此基础上,推广给出了二重积分中值定理逆命题的证明。

The famous first mean value for integrals is extremely significant in Mathematics Analysis,especially it is a important implement which by used in many subjects. So there are multiple improvements and generalizations. But, the inverse proposition of integral mean value theorems are not true generally. In this paper, we further discussed the inverse problem of the first mean value for integrals. Got the correct theorem and proofed it through improving the condition, On this base, we also prove the mean value theorem of the double integrals.