Lesson 2: 6.2 Approximating Areas with Riemann Sums

How can we estimate the area under a curve that doesn't form a perfect geometric shape?
What is the difference between a Left Riemann Sum and a Right Riemann Sum in terms of overestimation and underestimation?
How does increasing the number of subintervals (n) affect the accuracy of our approximation?
How do we translate the "sum of rectangles" into the formal language of calculus notation?

Riemann Sum
Subinterval
Partition
Delta
Left-endpoint Approximation
Right-endpoint Approximation
Midpoint
Riemann Sum
Sigma Notation
Limit of a Sum

College Board AP Calculus CED: CHA-4.B (Approximate a definite integral using geometric and numerical methods), LIM-5.A (Represent a definite integral as the limit of a Riemann sum).

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

This lesson moves from exact geometric areas to approximations of irregular curves. Students learn to divide an interval [a, b] into n subintervals and construct rectangles whose heights are determined by the function value at specific points. The lesson introduces the algebraic mechanics of finding Delta x and culminates in the transition from Sigma notation to the integral symbol, viewing the definite integral as an infinite sum of infinitely thin rectangles.

The purpose of 6.2 is to bridge the gap between basic geometry and the formal definition of integration. It builds the "Riemann Sum" foundation required for the AP Exam, where students are frequently asked to approximate values from tables of data rather than equations. It also reinforces the concept of limits, showing that calculus is the study of what happens as we move toward infinity.

DOK 1 (Recall): Calculating the width of subintervals for a given n.
DOK 2 (Skill/Concept): Computing a Riemann sum (Left, Right, or Midpoint) for a function over a specific interval.
DOK 3 (Strategic Thinking): Determining whether a Riemann sum is an over- or under-approximation based on whether the function is increasing or decreasing.

Support (Scaffolding):The "Rectangle Table": Provide a table with columns for i (rectangle number), the x-value used, the height, and Area to help students organize their calculations.
Visual Overlays: Use transparent graph paper or digital tools to physically draw rectangles over a curve so students can see the "error" (the gaps or overlaps).

Monotonicity Analysis: Ask students to prove why a Right Riemann sum always overestimates an increasing function regardless of the interval.
Sigma to Integral Translation: Provide complex limits of Riemann sums in Sigma notation and challenge students to identify the original function f(x) and the boundaries [a, b].Table of Values with Unequal Widths: Present a data table where the intervals between x-values are not uniform, forcing students to calculate each rectangle's area individually rather than factoring out a common Delta x.

AP College Board Assessments