Lesson 3: 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation

How can we use a finite number of rectangles to estimate the area of a region with a curved boundary?
What is the algebraic relationship between the number of rectangles (n), the width of each rectangle (Delta x), and the interval [a, b]?
As the number of rectangles approaches infinity, what happens to the approximation of the area?How do we "translate" the parts of a Sigma notation sum into the components of a definite integral?

Riemann Sum
Subinterval
Sigma Notation
Index of Summation
Lower/Upper Limits of Integration
Delta x
Integrand
Differential

College Board AP Calculus CED: LIM-5.A (Represent a definite integral as the limit of a Riemann sum), LIM-5.B (Calculate a definite integral using limits and summation notation).

Mathematical Practices: MP 4 (Communication and Notation), MP 1 (Implementing Mathematical Processes).

This lesson formalizes the "rectangle method" introduced earlier. Students learn to calculate Delta x and use it to find the heights of rectangles at specific x-values (Left, Right, or Midpoint).
The core of the lesson is the transition from the discrete sum to the continuous integral. Students practice identifying the function and the interval when presented with a limit of a Riemann sum, a common "reverse-engineering" task on the AP Exam.

The purpose of 6.3 is to establish the Definition of the Definite Integral. While 6.2 focused on the "why" of accumulation using simple geometry, 6.3 provides the "how" for any function. It is a critical conceptual bridge: it shows that integration isn't just "finding area," but a sophisticated limiting process of addition.

DOK 1 (Recall): Identifying the parts of the definite integral symbol (upper/lower limits, integrand).
DOK 2 (Skill/Concept): Computing a Left or Right Riemann sum for a given function and n value.
DOK 3 (Strategic Thinking): Converting a limit of a Riemann sum in Sigma notation into a definite integral, or vice versa.

Support (Scaffolding):Notation Match-Up: Create a "matching" activity where students pair a Sigma notation expression with its corresponding Definite Integral.
Step-by-Step Table: Use a template for Riemann Sums: Use a number line to show how the interval [a, b] is "cut" into n pieces before doing any algebra.
Extension (Enrichment):Generalizing the Sum: Ask students to write a Riemann sum for a function f(x) on [a, b] using n subintervals without being given a specific n, keeping everything in terms of variables.
Comparison Challenge: Given a strictly increasing function, ask students to rank the values of a Left Sum, a Right Sum, and a Midpoint Sum without calculating them.
Programmatic Thinking: Have students describe how they would write a simple computer loop to calculate a Riemann sum for n = 10,000.

AP College Board Assessments