# What is lower Riemann sum?

Jun 30, 2018

See below

#### Explanation:

The lower Riemann sum for the integral

${\int}_{a}^{b} f \left(x\right) \mathrm{dx}$

involves

• breaking up the interval $\left[a , b\right]$ into $N$ (not necessarily equal) pieces $\left[{x}_{0} , {x}_{1}\right) , \left[{x}_{1} , {x}_{2}\right) , \ldots , \left[{x}_{N - 1} , {x}_{N}\right]$ where ${x}_{0} = a$ and ${x}_{N} = b$
• evaluating the sum ${\sum}_{i = 1}^{N} {f}_{i} \left({x}_{i} - {x}_{i - 1}\right)$ where ${f}_{i}$ is the minimum value of $f \left(x\right)$ in the interval $\left[{x}_{i - 1} , {x}_{i}\right)$
• taking the limit of this sum so that the largest of the intervals go to zero ${\lim}_{\max \left\{{x}_{i} - {x}_{i - 1}\right\} \to 0} {\sum}_{i = 1}^{N} {f}_{i} \left({x}_{i} - {x}_{i - 1}\right)$

(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 $f \left(x\right)$ in the interval $\left[{x}_{i - 1} , {x}_{i}\right)$

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.