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The shell method slices the solid into thin cylindrical shells. Each has circumference 2πx, height f(x) and thickness dx, so the volume integrates 2πx·f(x).
Shell method
V = 2π ∫ₐᵇ x · f(x) dx
2π∫x³ dx = 2π × 4 = 8π ≈ 25.1327412.
Because the shells are centred on the y-axis, so their radius is the horizontal distance from that axis.