For questions on finding or constructing a partition of an interval to compute the Riemann integral or Riemann-Stieltjes integral.
A partition $\mathcal{P}$ of an interval $[a,b]$ is a sequence of points $t_0 = a < t_1 < t_2 < \dots < t_n = b$. Given a function $f : [a,b] \to \mathbb{R}$, the lower Riemann sum $L(f,\mathcal{P})$ is defined as $\sum_{i=1}^n m_{i}(f) (t_i - t_{i-1})$, where $m_i(f) = \inf \{ f(x) : x \in [t_{i-1},t_i] \}$. The upper Riemann sum $U(f,\mathcal{P})$ is similarly defined, with $M_i(f)$ in place of $m_i(f)$, where $M_i$ is defined by taking the supremum instead of infimum. A function is said to be Riemann integrable if $\sup_{\mathcal{P}} L(f,\mathcal{P}) = \inf_\mathcal{P} U(f,\mathcal{P})$.