The Cubic Horn

y = x−3 revolved about the x-axis over [1, ∞)

∞
Volume
0.628
limit π/5 ≈ 0.6283
Lateral surface
4.863
limit ≈ 4.8628

The two equations

Graph relation (read from both screenshots):

1 / y = x3  ⟷  y = x−3

Differential form (Δ moved out of the ΣΔ label):

Δy / Δx = −3 x2 y2

Volume & surface area

V = π ∫1∞ x−6 dx = π/5 ≈ 0.6283
S = 2π ∫1∞ x−3√(1 + (3x−4)2) dx ≈ 4.8628

Unlike the classic Gabriel’s horn (y = 1/x), the steeper cube decay makes both the volume and the surface area finite.

The exponent 3 is fixed by the graphs (odd symmetry, blow-up at zero, decay at infinity). Volume and surface values are exact calculus on y = x−3. The revolution axis and the interval [1, ∞) are modelling choices — x = 1 is the curves’ crossing point and x = 0 is an infinite singularity that cannot anchor a solid.