ABSTRACT
Interestingly, the more advanced idea of the Lebesgue integral is obtained by breaking up the range of the function. This chapter presents calculus that it is often useful to think of an integral as representing area. However, this is but one of many important applications of integration theory. The integral is a generalization of the summation process. That is the point of view taken in the present chapter. The first main step in the theory of the Riemann integral is to determine a method for "calculating the limit of the Riemann sums" of a function as the mesh of the partitions tends to zero. The value of the integral is well approximated by its Riemann sums. This observation is a useful tool in calculation. Of course the integral is a linear operator on functions, and enjoys thereby a number of useful properties. These include ways in which the integral respects arithmetic operations.
