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Grade 12 Math Lab: Analysis & Inference FREE
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Solids of Revolution
Learning Objectives
- Use disk method
- Use washer method
- Use shell method
Key Concepts
Disk Method
V = π∫[a,b] [f(x)]² dx
Shell Method
V = 2π∫[a,b] x·f(x) dx
Theory
**Disk Method** (rotate about x-axis):
V = π∫[a,b] [f(x)]² dx
**Washer Method** (hollow solid):
V = π∫[a,b] ([R(x)]² - [r(x)]²) dx
**Shell Method** (rotate about y-axis):
V = 2π∫[a,b] x·f(x) dx
**Choose method:** Disk/Washer when slicing ⊥ to axis. Shell when slicing ∥ to axis.
Examples
Rotate y = x² from 0 to 2 about x-axis
Solution: V = π∫₀² (x²)² dx = π∫₀² x⁴ dx = π[x⁵/5]₀² = 32π/5