Riemann and Darboux Sums

Abstract

Let $f: [a,b] \rightarrow \mathbb{R}$ be continuous and positive. By the subgraph of f, we mean the region from the xy-plane delimited by the x-axis, the lines x = a, x = b, and the curve y = f(x). More precisely, the subgraph is the set

$$\displaystyle{\left \{(x,y) \in \mathbb{R}^{2}\;\vert \;a \leq x \leq b,\;0 \leq y \leq f(x)\right \}.}$$