Cross-sections, solids of revolution, and Cavalieri's Principle
• Identify two-dimensional cross-sections of three-dimensional objects. • Identify three-dimensional objects generated from rotating two-dimensional shapes. • Use Cavalieri’s Principle to obtain the formula for the volume of a pyramid from the formula for the volume of a pyramid with square base. • Use Cavalieri’s Principle to obtain the formula for the volume of a cone.
In this unit students give more sophisticated, although still informal, arguments for the circumference, area, and volume formulas that they learned in earlier grades. These arguments rely on dissections, informal limits, and Cavalieri’s Principle. After giving arguments for the volume formulas, students use these formulas in solving a variety of modeling problems, including problems that involve density.
Tasks
1
Tennis Balls in a Can
2
Use Cavalieri’s Principle to Compare Aquarium Volumes