> [!NOTE] Definition (Lower Riemann Sum) > Let $[a,b]$ be a [[Bounded Real Function|bounded]] [[Closed Real Interval|closed real interval]]. Let $f:[a,b]\to \mathbb{R}$ be a [[Real Function|real function]]. Let $P=\{ x_{0},x_{1},\dots,x_{n} \}$ be a [[Finite Partition of Closed Real Interval|finite partition]] of $[a,b].$ For $i=1,\dots,n,$ let $m_{i}$ be the [[Infimum of Set of Real Numbers|infimum]] of the [[Image of a set under a function|image]] of $[x_{i-1},x_{i}]$ under $f.$ Then lower Riemann sum of $f$ with respect to $P$ is given by $L(f,P)=\sum_{i=1}^{n} m_{i}(x_{i}-x_{i-1})$ # Properties ... # Applications ...