Applications of Integrals · Volume
Shell Method
Volume of revolution using cylindrical shells. Useful when revolving about the y-axis or when the disk/washer method leads to difficult integrals.
Conditions. As written, the region under y = f(x) on [a, b] is revolved about the y-axis with 0 ≤ a < b, so the shell radius is x itself. For a different axis of revolution x = c, replace the radius x with |x - c|.
- 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 y = x² on [0, 1] is revolved about the y-axis.
- V = 2π ∫₀¹ x · x² dx = 2π ∫₀¹ x³ dx = 2π[x⁴/4]₀¹ = 2π/4 = π/2
Answer: π/2 ≈ 1.5708
1 more worked examplePremium
Find the volume when the region under y = √x on [0, 4] is revolved about the y-axis.
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Practice
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