Question

What is lower Riemann sum?

Answer 1

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.