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Revolving about the x-axis stacks discs of radius f(x), giving V = π∫f(x)² dx. Revolving about the y-axis sweeps out cylindrical shells of radius x and height f(x), giving V = 2π∫x·f(x) dx.
Disc method
V = π ∫ₐᵇ [f(x)]² dx
Shell method
V = 2π ∫ₐᵇ x·f(x) dx
π∫x² dx = 8π/3 ≈ 8.3775804, the cone of radius 2 and height 2.
Whichever avoids solving for the other variable. Discs suit rotation about the axis of the variable you integrate over.