Two Methods of Proving the Improved Mean Value Theorem of Integral

The proof of the mean value theorem for integral, which is given by Advanced Mathematics and which is wildly used, only proved that the mean value  is on the closed interval   b a, . In this paper, we provide two different methods for the proof of the mean value theorem of integral and prove the mean value  is in the open interval   b a, , which is an improvement in the conclusion of the theorem. In the end, we illuminate the practicability of the improved mean value theorem for integral with two examples as follows.