Integrating The Concept Of Modern Development

The concept of integral is one of the most important concepts in the modernanalytic mathematics. When calculous was at it's initial stage, Newton computed the integral by contradictorily operation of differential, Leibniz used the "tiny elements" method. 1823,Cauchy defined integral the "limit of sum", he then gave the structural definition of definite integral for continuous functions. In the middle of 19th century , Riemann introduced Riemann integral and discovered the essential of definite integral profoundly and completely . The integrability theory was improved substantively by Darboux at the end of 19th century. Later on , Stieltjes introduced a new integral to show the limit of analytic function sequence , and this became the outset of integral on general measure . At the beginning of 20* century ,Lebesgue rebuilt Riemann integral and introduced Lebesgue integral . With the development of modern mathematics, the concept of integral develops too. This thesis mainly focuses on the integral of Newton, Leibinz, Cauchy, Riemann, Lebesgue, Stieltjes and the other forms in modem times. The author puts stress on analysis how the modem integral notion develops according to the need of theory and application.