The left hand rule for computing the Riemann integral $\displaystyle\int\limits_{a}^{b}f(x)\,dx$ is

If the Riemann integral is considered as a measure of area under a curve, then the expressions $\displaystyle f\left(a+(j-1)\left(\frac{b-a}{n}\right)\right)$ represent the heights of the rectangles, and $\displaystyle\frac{b-a}{n}$ is the common width of the rectangles.

The Riemann integral can be approximated by using a definite value for $n$ rather than taking a limit. In this case, the partition is $\displaystyle\left\{\left[a,a+\frac{b-a}{n}\right),\dots,\left[a+\frac{(b-a)(n-1)}{n},b\right]\right\}$, and the function is evaluated at the left endpoints of each of these intervals. Note that this is a special case of a left Riemann sum in which the $x_{j}$’s are evenly spaced.