Integral method for approximating sums

We wish to approximate a sum $S_n = \sum_{k=1}^n f(k)$. If $f(k)$ is either weakly increasing or weakly decreasing over some interval $[1,n]$ we can approximate it using an integral.

Let $I_n = \int_{x=0}^n f(x) dx$.

If $f(x)$ is weakly increasing over $[1,n]$ we can say $I_n + f(1) \le S_n \le I_n + f(n)$.

If $f(x)$ is weakly decreasing over $[1,n]$ we can say $I_n + f(1) \ge S_n \ge I_n + f(n)$.