Riemann sum
Let be a closed interval, be bounded on , , and be a partition of . The Riemann sum of over with respect to the partition is defined as
where is chosen arbitrary.
If for all , then is called a left Riemann sum.
If for all , then is called a Riemann sum.
Equivalently, the Riemann sum can be defined as
where is chosen arbitrarily.
If , then is called an upper Riemann sum.
If , then is called a lower Riemann sum.
For some examples of Riemann sums, see the entry examples of estimating a Riemann integral.
Title | Riemann sum |
Canonical name | RiemannSum |
Date of creation | 2013-03-22 11:49:17 |
Last modified on | 2013-03-22 11:49:17 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 14 |
Author | Wkbj79 (1863) |
Entry type | Definition |
Classification | msc 26A42 |
Related topic | RiemannIntegral |
Related topic | RiemannStieltjesIntegral |
Related topic | LeftHandRule |
Related topic | RightHandRule |
Related topic | MidpointRule |
Defines | left Riemann sum |
Defines | right Riemann sum |
Defines | upper Riemann sum |
Defines | lower Riemann sum |