What is lower Riemann sum?
1 Answer
Jun 30, 2018
See below
Explanation:
The lower Riemann sum for the integral
involves
- breaking up the interval
#[a,b]# into#N# (not necessarily equal) pieces#[x_0,x_1),[x_1,x_2),... ,[x_{N-1},x_N]# where#x_0=a# and#x_N=b# - evaluating the sum
#sum_{i=1}^N f_i (x_i-x_{i-1})# where#f_i# is the minimum value of#f(x)# in the interval#[x_{i-1},x_i)# - taking the limit of this sum so that the largest of the intervals go to zero
#lim_{max{x_i-x_{i-1}} to 0} sum_{i=1}^N f_i (x_i-x_{i-1})#
(some authors call the result of the summation the lower Riemann sum, the limit being called the lower Riemann integral)
We can define the upper Riemann sum in a similar fashion - but with
For the integral to exist, both the lower and the upper Riemann sum must exist and be equal - their common value is the Riemann integral.
