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The common value determined by Riemann sums when a function is integrable.
A Riemann integral of a Riemann integrable function is the number
defined by
where and are the lower and upper sums over all partitions of .
This definition matches the limit of Riemann sums along partitions of small mesh, and it is the starting point for results such as linearity and the Fundamental Theorem of Calculus.
A lower sum of a bounded function with respect to a partition is the number
where is the infimum of on the th subinterval.
Lower sums are paired with upper sums to define Riemann integrability via the gap .
A Riemann sum of a bounded function with respect to a tagged partition , where , is the number
Riemann sums approximate the Riemann integral; the approximation is controlled by the mesh becoming small.
Riemann linearity: Let be Riemann integrable on the interval , and let . Then is Riemann integrable on , and
This is the basic linear structure behind the Riemann integral and is used throughout the algebraic manipulation of integrals.
Fundamental theorem of calculus I: Let and let be a Riemann integrable function. Define
Then is continuous on . Moreover, if is continuous at a point , then is differentiable at and