Riemann Sums

Let $f$ be defined on $[a,b]$ and let $\Delta$ be a partition of $[a,b]$ given by $a=x_0<x_1<x_2<\cdots<x_{n-1}<x_n=b,$ where $\Delta x_i$ is the length of the $i$ th subinterval. If $c_i$ is any point in the $i$ th subinterval, then the sum

$$\sum_{i=1}^n f(c_i) \Delta x_i,\qquad x_{i-1}\le c_i\le x_i$$

is called a Riemann Sum of $f$ for the partition $\Delta.$ The length of the largest subinterval of a partition $\Delta$ is called the norm of the partition, and is denoted $\left\lVert\Delta \right\rVert .$