Riemann sum
In
mathematics, a
Riemann sum is a method for approximating the values of integrals.
Let it be supposed there is a function f: D → R where D, R &sube R and that there is a closed interval I = [a,b] such that I &sube D. If we have a finite set of points {x0, x1, x2, ... xn} such that a = x0 < x1 < x2 ... < xn = b, then this set creates a partition P = {[x0, x1), [x1, x2), ... [xn-1, xn]} of I.
If is a partition with elements of , then the Riemann sum of over with the partition is defined as
where x
i-1 ≤ y
i ≤ x
i. The choice of y
i is arbitrary. If y
i = x_
i-1 for all
i, then
S is called a
left Riemann sum. If y
i = x
i, then
S is called a
right Riemann sum.
Suppose we have
where
vi is the
supremum of
f over [x
i-1, x
i]; then
S is defined to be an
upper Riemann sum. Similarly, if
vi is the
infimum of
f over [x
i-1, x
i], then
S is a
lower Riemann sum.
See also: