Bounded interval/Step function/Step integral/Definition

Step integral

Let $I$ be a real interval with endpoints $a, b \in ℝ$ , and let

t : I ⟶ ℝ

denote a step function for the partition $a = a_{0} < a_{1} < a_{2} < \dots < a_{n - 1} < a_{n} = b$ , with the values $t_{i}$ , $i = 1, \dots, n$ . Then

T := \sum_{i = 1}^{n} t_{i} (a_{i} - a_{i - 1})

is called the step integral of $t$ on $I$ .