A trumpet-shaped solid holds a finite bucket of paint but has infinite surface area
Take the curve y = 1/x for x >= 1 and spin it around the x-axis to form an infinitely long, trumpet-shaped solid, narrowing forever but never quite reaching zero width. Can a shape with infinite surface area really hold only a finite volume inside it, and if so, why couldn't that same finite paint coat its own infinite surface?
Reveal the answer
Yes, the volume is finite (exactly pi cubic units) while the surface area is infinite, a result Evangelista Torricelli proved in 1641, decades before calculus was formally developed, provoking fierce objections at the time. The seeming contradiction, sometimes called the 'painter's paradox,' dissolves once you realize that filling the horn with a finite volume of paint does coat its entire infinite inner surface, since the paint layer simply grows thinner and thinner as the horn narrows, approaching zero thickness without needing infinite material.
— Evangelista Torricelli, De solido hyperbolico acuto — 1641