# Riemann Integral

(redirected from*Riemann integrable*)

## Riemann integral

[′rē‚män ‚int·ə·grəl] (mathematics)

The Riemann integral of a real function ƒ(

*x*) on an interval (*a,b*) is the unique limit (when it exists) of the sum of ƒ(*a*_{i })(*x*_{i }-*x*_{i-1}),*i*= 1, …,*n*, taken over all partitions of (*a,b*),*a*=*x*_{0}<>*a*_{1}<>*x*_{1}< ⋯=""><>*a*_{n }<>*x*_{n }=*b*, as the maximum distance between*x*_{i }and*x*_{i-1}tends to zero.## Riemann Integral

the usual definite integral. The definition of the Riemann integral was in essence given by A. Cauchy in 1823. Cauchy, however, applied his definition to continuous functions. In a paper written in 1853 and published in 1867, B. Riemann was the first to point out the necessary and sufficient conditions for the existence of the definite integral. In modern terms, these conditions can be stated as follows: for the definite integral of a function to exist over some interval, it is necessary and sufficient that (1) the interval be finite, (2) the function to be bounded on the interval, and (3) the set of discontinuities of the function on the interval have Lebesgue measure zero (*see*MEASURE OF A SET).