Riemann lower/ upper sum

Overview

Can be seen as an approximation to the integral of a function. Works by cutting the integral up into finite small pieces and making a square under the function’s graph.

Definition

Lower sum: $L(f,P)={\sum }_{i=1}^n(x_i-x_{i-1}){\inf }_{[x_{i-1},x_i]}f$

Upper sum: $U(f,P)={\sum }_{i=1}^n(x_i-x_{i-1}){\sup }_{[x_{i-1},x_i]}f$

Note that the upper sum will always be an over estimation of the integral and the lower sum will always be an under estimation.

The lower Riemann integral is ${\int }_{\_}f={\sup }_{P}L(f,P)$

and the upper Riemann integral is ${\int }^{\_}f={\inf }_{P}U(f,P)$