information:extraction
Riemann integral

The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. A tagged partition of a closed interval on the real line is a finite sequence

This partitions the interval into n sub-intervals indexed by i, each of which is "tagged" with a specific point t ∈. A Riemann sum of a function f with respect to such a tagged partition is defined as

thus each term of the sum is the area of a rectangle with height equal to the function value at the chosen point of the given sub-interval, and width the same as the width of sub-interval, Δ = x − x. The mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, max Δ. The Riemann integral of a function f over the interval is equal to S if:

When the chosen tags are the maximum (respectively, minimum) value of the function in each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum, suggesting the close connection between the Riemann integral and the Darboux integral.