How to Find the Volume and Surface Area of Gabriel's Horn

Finding the volume and surface area of this horn problem may blow your mind. Gabriel’s horn is the solid generated by revolving about the x-axis the unbounded region between

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This figure shows a graph of Gabriel's horn.

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Playing this instrument poses several not-insignificant challenges: 1) It has no end for you to put in your mouth; 2) Even if it did, it would take you till the end of time to reach the end; 3) Even if you could reach the end and put it in your mouth, you couldn’t force any air through it because the hole is infinitely small; 4) Even if you could blow the horn, it’d be kind of pointless because it would take an infinite amount of time for the sound to come out. There are additional difficulties — infinite weight, doesn’t fit in universe, and so on — but you get the picture.

Believe it or not, despite the fact that Gabriel’s horn has a finite volume, it has an infinite surface area!

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You find the total volume by adding up the little bits from 1 to infinity.

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So, the total volume of this infinitely long trumpet is, roughly, a measly 3.14 cubic units. To determine the surface area, you first need the function’s derivative:

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Now plug everything into the surface area formula.

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This is an improper integral, so when you solve it, you determine that

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Bonus question for those with a philosophical bent: Assuming Gabriel is omnipotent, could he overcome the above-mentioned difficulties and blow this horn? Hint: All the calculus in the world won’t help you with this one.

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