$\sum _{k\=1}^{n}f\left(x_k^{\ast }\right){\mathrm{\Δ}x}_k$

where $x_0\=a$, $x_n\=b$, and $\left\{x_k\right\}\,k\=1\,\dots \,n-1$, form a partition of the interval $\left[a\,b\right]$, with ${\mathrm{\Δ}x}_k\=x_k-x_{k-1}$, and $x_k^{\ast }\in \left[x_{k-1}\,x_k\right]$.

upper sum

$x_k^{\ast }$ are chosen so that each $f\left(x_k^{\ast }\right)$ is a maximum on $\left[x_{k-1}\,x_k\right]$

sum is a maximum for that partition

lower sum

$x_k^{\ast }$ are chosen so that each $f\left(x_k^{\ast }\right)$ is a minimum on $\left[x_{k-1}\,x_k\right]$

sum is a minimum for that partition

midpoint sum

$x_k^{\ast }$ are chosen as the midpoints of each subinterval $\left[x_{k-1}\,x_k\right]$

left sum

$x_k^{\ast }\=x_{k-1}$ (evaluate $f$ at the left end of each subinterval)

right sum

$x_k^{\ast }\=x_k$      (evaluate $f$ at the right end of each subinterval)

random sum

$x_k^{\ast }$ are selected at random in each subinterval $\left[x_{k-1}\,x_k\right]$

Table 4.1.1   Six types of Riemann sums

Example 4.1.1

Use a left Riemann sum to obtain the area bounded by the graph of $f\left(x\right)\=6\+x-x^2$ and the $x$-axis.

Example 4.1.2

Use a Riemann sum to obtain the area bounded by the graph of $f\left(x\right)\=x^3-2x^2-5 x\+6$ and the $x$-axis.

Example 4.1.3