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Torricelli's Trumpet

Torricelli's trumpet is a geometric figure formed by rotating the fuction y = 1/x around the x axis. You start at x = 1 and then go on forever.

Torricelli's Trumpet

Using elementary calculus you can prove the volume of Torricelli's trumpet is equal to π and yet the area is infinite. Ultimately this apparent paradox is due to the dimensions of area as compared to volume. In a nutshell, the volume - being a cubic function - converges faster than the area, which is a squared function.

Attempts to rationalize the phenomenon by comparing it to the real world are often correct but usually miss the point. For instance, you can't paint the inside of the trumpet, some say, because eventually the diameter is so thin the paint won't flow in. However, even without that restriction you can still paint the whole infinite area of Torricelli's trumpet as long as you define "paint" to mean covering a surface with a finite thickness - however, thin - of a fluid. But you have to paint it just right.

In fact, you can paint the whole infinite area of Torricelli's trumpet with as small amount as you like. In other words, the whole interior of a Torricelli's trumpet which starts with a diameter of 20 miles - and can hold 3,459,253,777,669,599 gallons of paint - can be painted with a single drop. The proof of this is quite simple, but for now it will be left - as the mathematics textbooks say - as an exercise for the reader.