Every day, traders compute the P&L of a strategy over a time window. If the P&L rate (in ) varies continuously, the total P&L is the area under the rate curve — a geometric problem. Every pricing formula involving an expectation over a continuous distribution is, in the end, an area under a probability density. The common structure behind both is the Riemann sum: slice the domain into thin strips, approximate each strip's contribution by a rectangle, and take the limit as the strips become infinitely thin. The result is the definite integral.
This note builds the Riemann sum from scratch — from estimating area with four rectangles to the formal sigma-notation definition — and shows how the same limit resolves the distance problem (total displacement from variable velocity). The Riemann sum is the meaning of the integral; the computational shortcut (antiderivatives via the Fundamental Theorem) is the subject of the next two lessons.
Learning objectives. By the end of this note you should be able to:
- compute left-, right-, and midpoint-endpoint Riemann sums for a given function and partition;
- write a Riemann sum in sigma notation and simplify using standard summation formulas;
- state the definition of the definite integral as a limit of Riemann sums;
- apply the distance/displacement interpretation: ;
- explain why the limit exists for continuous functions (Riemann integrability).
In this lesson
- The area problem
- Sigma notation and summation formulas
- The definite integral: the limit of Riemann sums
- The distance problem
- Riemann sums in finance: a numerical illustration
Before you start
- The Limit of a Function: Intuition, One-Sided Limits, and Notation
- The Derivative as a Limit: Rates of Change and the Derivative Function
The full lesson, with worked examples and exercise solutions, is for members.
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