What is lower Riemann sum?

1 Answer
Jun 30, 2018

See below

Explanation:

The lower Riemann sum for the integral

# int_a^b f(x) dx #

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 #f_i# standing for the maximum value of the function #f(x)# in the interval #[x_{i-1},x_i)#

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.