Applications of Integrals · Volume
Washer Method
Volume of revolution when there is a gap between the curve and the axis. R(x) is the outer radius and r(x) is the inner radius.
- 2 variables
- 2 worked examples
- 14 in Applications of Integrals
Variables
| Symbol | Name | Unit |
|---|---|---|
| a | Left bound | - |
| b | Right bound | - |
Worked examples
Find the volume when the region between y = x² and y = x is revolved about the x-axis.
- Intersection: x² = x → x = 0, 1. On [0,1]: R = x, r = x²
- V = π ∫₀¹ (x² - x⁴) dx = π[x³/3 - x⁵/5]₀¹ = π(1/3 - 1/5) = 2π/15
Answer: 2π/15 ≈ 0.4189
1 more worked examplePremium
Find the volume when the region under y = x on [0, 1] is revolved about the line y = -1.
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Practice
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