Upper and lower Darboux sums

Definition

Let $a < b$, let $f \colon [a, b] \to \mathbb{R}$ be bounded, and let $P = (x_0, \dots, x_n)$ be a partition of $[a, b]$. For each $i$, let $M_i = \sup f([x_{i-1}, x_i])$ and $m_i = \inf f([x_{i-1}, x_i])$. The upper and lower Darboux sums of $f$ for $P$ are $$U(f, P) = \sum_{i=1}^{n} M_i (x_i - x_{i-1}), \qquad L(f, P) = \sum_{i=1}^{n} m_i (x_i - x_{i-1}).$$

Remarks

Always $(b - a) \inf f \le L(f, P) \le U(f, P) \le (b - a) \sup f$. For a refinement $Q$ of $P$, we have $L(f, P) \le L(f, Q)$ and $U(f, Q) \le U(f, P)$. So $L(f, P) \le U(f, Q)$ for any two partitions $P$ and $Q$.

Explanations

L1 human

The upper sum covers the graph with rectangles that reach its highest point on each piece. The lower sum uses rectangles that reach its lowest point on each piece.