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
- 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
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Common questions
What is the 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. It is one of the applications of integrals formulas in the CalcRef reference.
What do the symbols in the Washer Method mean?
In the Washer Method, a is the left bound; b is the right bound.
How do you use the Washer Method?
Worked example. 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.
Practice
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