The definite integral is a number — the limit of Riemann sums — and it represents net signed area: positive where is above the -axis, negative where it is below. In finance, the same integral represents accumulated net cash flow: inflows add, outflows subtract. The properties of the integral (linearity, additivity, comparison) are exactly the properties you would expect from signed area, and they allow an analyst to decompose a complex integral into simpler pieces.
This note formalises Definition 2 from Stewart, states and proves the key properties, and introduces the geometric interpretation of negative-area regions. The Fundamental Theorem of Calculus — the practical tool for computing these integrals — is the subject of the next lesson.
Learning objectives. By the end of this note you should be able to:
- state the formal definition of the definite integral as a limit of Riemann sums;
- interpret the definite integral as signed area when takes negative values;
- apply the seven properties (including linearity, additivity, and comparison) to simplify integrals;
- evaluate simple definite integrals using properties alone (constant, linear functions);
- recognise that is a number, not a function.
In this lesson
- The formal definition
- Signed area
- Properties of the definite integral
- Using properties to evaluate integrals
- Evaluating integrals via the limit: a worked example
Before you start
- Areas and Distances: The Riemann Sum
- The Limit of a Function: Intuition, One-Sided Limits, and Notation
The full lesson, with worked examples and exercise solutions, is for members.
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