Problems And Counterexamples In Riemann Integral

In the early 20th century, the development of Lebesgue measure and integral settle a foundation for latest mathematic analysis. However the base of this change and development was classical Riemann integral. Thus the concept and theory of Riemann integral is very important. During mathematical analysis teaching procedure, Riemann integral occupies the leading position. Meanwhile, it's the important continuous course such as constant differential equation, complex function, real function theory, probability theory, dynamical course and so on after mathematical analysis.The papers mainly analyzed the mistake appearing in the process of integral calculation and integral proving in the advanced math and math analysis and then made a conclusion of how to deal with these problems correctly. The fact proved that understanding Riemenn integral concept as well as its character is the key point.The paper consists of the following six parts:Chapter one introduced the background, motivation and significance of choosing the title.Chapter two gave out some relative concepts, characters and calculation methods of indefinite integral,definite integral,the second curved area integral.Chapter three displayed the common mistakes appearing in the indefinite integral of the advanced math.Chapter four made the conclusion of common mistakes turning out during the procedure of calculating and proving the definite integral.Chapter five analyzed the mistakes in calculating the second curved area integral and the problems should be taken care of.Chapter six made a little conclusion to the mistakes easily coming out in Riemann integral, meanwhile it gave out the base of understanding Riemann integral concepts.