CalcRef

Applications of Integrals · Volume

Shell Method

V=2π∫abx⋅f(x) dxV = 2\pi \int_a^b x \cdot f(x) \, dx

Volume of revolution using cylindrical shells. Useful when revolving about the y-axis or when the disk/washer method leads to difficult integrals.

Variables

SymbolNameUnit
aLeft bound—
bRight bound—

Worked examples

Find the volume when y = x² on [0, 1] is revolved about the y-axis.
  1. 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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