Unit 13 · Lesson 02 · Week 35 · Applications of Integration & Calculus Capstone
Volumes with Known Cross Sections
Integral applications begin with geometry and slicing. The variable of integration determines slice orientation; every area or volume element must be expressed consistently in that variable. This lesson emphasizes volumes with known cross sections through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in volumes with known cross sections.
- 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 derivatives, integrals, and differential equations support a defensible analysis of a complex problem?
Visual model
Integral applications begin with geometry and slicing. The variable of integration determines slice orientation; every area or volume element must be expressed consistently in that variable.
A=\int_a^b(\text{top}-\text{bottom})dxV=\int_a^b A(x)dxV=\pi\int_a^b(R^2-r^2)dx
Worked reasoning
Misconception check: For area between curves, the integrand must remain nonnegative on the selected interval or the interval must be split.
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
Volumes with Known Cross Sections
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Find the area between y=x and y=x² on [0,1].
Answer: 1/6
Reasoning: Integrate x−x² from 0 to 1.
Reasoning: Integrate x−x² from 0 to 1.
Write a washer integral with outer radius R and inner radius r.
Answer: π∫(R²−r²)dx
Reasoning: Washer area is π(R²−r²).
Reasoning: Washer area is π(R²−r²).
A solid has square cross sections of side x on [0,2]. Find its volume.
Answer: 8/3
Reasoning: Integrate x² from 0 to 2.
Reasoning: Integrate x² from 0 to 2.
Find the average value of f(x)=x on [0,4].
Answer: 2
Reasoning: (1/4)∫₀⁴x dx=2.
Reasoning: (1/4)∫₀⁴x dx=2.
Why sketch before integrating?
Answer: It identifies bounds, orientation, and positive dimensions.
Reasoning: The geometry determines the integrand.
Reasoning: The geometry determines the integrand.
When must area-between-curves intervals be split?
Answer: When the top and bottom curves switch.
Reasoning: The geometric height must remain nonnegative.
Reasoning: The geometric height must remain nonnegative.