Riemann sum

A Riemann sum is a finite approximation to the Riemann Integral.

Definition

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n$ be a tagged partition on $[a,b]$

The Riemann sum of $f$ with respect to $\mathcal{\dot{P}}$ on $[a,b]$ is defined as $S(f,\mathcal{\dot{P}})=\sum_{i=1}^n f(t_i)(x_i-x_{i-1})$

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $\mathcal{P}=\{[x_{i-1},x_i]\}_{i=1}^n$ be a partition on $[a,b]$

Let $M_i=\sup \{f(x):x\in [x_{i-1},x_i]\}\forall i$

The Upper sum of $f$ with respect to $\mathcal{P}$ on $[a,b]$ is defined as $U(f,\mathcal{P})=\sum_{i=1}^n M_i (x_i-x_{i-1})$

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $\mathcal{P}=\{[x_{i-1},x_i]\}_{i=1}^n$ be a partition on $[a,b]$

Let $m_i=\inf \{f(x):x\in [x_{i-1},x_i]\}\forall i$

The Lower sum of $f$ with respect to $\mathcal{P}$ on $[a,b]$ is defined as $L(f,\mathcal{P})=\sum_{i=1}^n m_i (x_i-x_{i-1})$