Unit 09 · Lesson 03 · Week 25 · Accumulation, Riemann Sums & Definite Integrals
Midpoint Sums
Riemann sums approximate accumulation by adding height-times-width contributions. The sample-point rule and partition determine the estimate; refinement generally improves resolution. This lesson emphasizes midpoint sums through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in midpoint sums.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How can infinitely refined sums measure net change and accumulated quantity?
Visual model
Riemann sums approximate accumulation by adding height-times-width contributions. The sample-point rule and partition determine the estimate; refinement generally improves resolution.
\sum_{i=1}^n f(x_i^*)\Delta x\Delta x=\frac{b-a}{n}
Worked reasoning
Misconception check: The number of partition points and the number of subintervals are not the same.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Midpoint Sums
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
For [0,4] with n=4, find Δx.
Answer: 1
Reasoning: (4−0)/4=1.
Reasoning: (4−0)/4=1.
Use a left sum for f(x)=x² on [0,2] with n=2.
Answer: 1
Reasoning: Widths are 1 and left heights are 0 and 1.
Reasoning: Widths are 1 and left heights are 0 and 1.
Use a right sum for the same partition.
Answer: 5
Reasoning: Right heights are 1 and 4.
Reasoning: Right heights are 1 and 4.
Use a midpoint sum for f(x)=x on [0,2] with n=2.
Answer: 2
Reasoning: Midpoints 0.5 and 1.5 sum to 2 with width 1.
Reasoning: Midpoints 0.5 and 1.5 sum to 2 with width 1.
For an increasing function, which is typically larger: left or right sum?
Answer: Right sum
Reasoning: Right endpoints have larger heights.
Reasoning: Right endpoints have larger heights.
How does refinement affect a Riemann estimate?
Answer: Smaller widths usually improve resolution.
Reasoning: The rectangles follow the curve more closely.
Reasoning: The rectangles follow the curve more closely.