Riemann Sum

Let

• $f\,:\,[a,b]\rightarrow\mathbb{R}$ is a bounded function on the compact interval $I=[a,b]$ with $M=\sup_I f$ and $m=\inf_I f$.
• $P=\set{I_1, I_2, \dots, I_n}$
• $M_k=\sup_{I_k} f = \sup\set{f(x):x\in[x_{k-1},x_k]}$
• $m_k=\inf_{I_k} f = \inf\set{f(x):x\in[x_{k-1},x_k]}$

Upper riemann sum

$$U(f;P)=\sum_{k=1}^{n}{M_k\lvert{I_k}\rvert}$$

Lower riemann sum

$$L(f;P)=\sum_{k=1}^{n}{m_k\lvert{I_k}\rvert}$$

$m_k<M_k \implies L(f;P)\,\le\,U(f;P)$ $$

When $P_1, P_2$ are any 2 partitions of $I$: $L(f;P_1) \le U(f;P_2)$