Riemann sum
Let I=[a,b] be a closed interval, f:I→ℝ be bounded
on I, n∈ℕ, and P={[x0,x1),[x1,x2),…[xn-1,xn]} be a partition of I. The Riemann sum
of f over I with respect to the partition P is defined as
S=n∑j=1f(cj)(xj-xj-1) |
where cj∈[xj-1,xj] is chosen arbitrary.
If cj=xj-1 for all j, then S is called a left Riemann sum.
If cj=xj for all j, then S is called a Riemann sum.
Equivalently, the Riemann sum can be defined as
S=n∑j=1bj(xj-xj-1) |
where bj∈{f(x):x∈[xj-1,xj]} is chosen arbitrarily.
If bj=sup, 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 |