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.
- 2 variables
- 2 worked examples
- 14 in Applications of Integrals
Variables
| Symbol | Name | Unit |
|---|---|---|
| a | Left bound | - |
| b | Right bound | - |
Worked examples
- V = 2π ∫₀¹ x · x² dx = 2π ∫₀¹ x³ dx = 2π[x⁴/4]₀¹ = 2π/4 = π/2
Answer: π/2 ≈ 1.5708
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Common questions
What is the 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. It is one of the applications of integrals formulas in the CalcRef reference.
When does the Shell Method apply?
The Shell Method holds under this condition: 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|.
How do you use the Shell Method?
Worked example. 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.
Practice
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