Volume of Gabriel's Horn

Calculus 2 · Axiom Academy

Gabriel's Horn: The Painter's Paradox Spin one humble curve, y = 1/x, and you build a trumpet you can fill with paint but never paint. Take the curve y = 1/x for x 1 and revolve it about the x-axis. The horn that forms runs forever , yet — as you're about to see — it holds only a finite puddle of paint. Drag the cutoff and watch the horn build: each slice is the height of the actual y = 1/x curve, spun into a circle. Push it out and the horn stretches and thins toward zero — but never closes off. Pour in the paint — the volume stops at π Fill the horn out to x = b. The disk method gives V = π(1 − 1/b). Drag b out toward infinity and the fill bar does something surprising: it rises up to the π ceiling and stalls just below it — refusing to go further. Now paint the wall — the surface never stops growing Same horn, same cutoff b — but stack volume against the surface area. The surface is at least 2π·ln b, and ln b has no ceiling. Drag b and watch the volume bar freeze while the surface bar runs off the chart. The painter's paradox. You could pour exactly π cubic units of paint to fill Gabriel's Horn, yet its inner wall has infinite area — so no finite coat of paint could ever cover it. Finite to fill, infinite to paint.

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