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Gabriel’s horn

gabriels horn

Gabriel’s horn or Torricelli’s trumpet is the surface of revolution of the function $ f(x) = \frac{1}{x}$ about the x – axis for $ x \ge 1$. What is this exactly? First draw your axes and draw function $\frac{1}{x}$ for $ x \ge 1$.

gabriel's horn graph

Now you imagine it rotating around x – axis.

rotating graph around x-axis

gabriels horn

Image credit to Fouriest Series

This figure has finite volume but infinite surface area.

How is this possible?

Well let’s try to find out what’s going on by calculation. This requires simple integral calculations. If we cut this horn into tiny regular slices we’ll always get circles with radius $\frac{1}{x}$ which means that volume of one little slice is equal to $\frac{1}{x^2} \pi$. And now that we know that we can find the volume of whole horn. Since the upper border of integral is infinity we have to use limit to get what we want.

volume formula

This means that the volume of this trumpet is equal to π cubic units.

To calculate surface we’ll use surface integrals of second kind.

surface formula

animation gabriel horn

What confused people for a long time is a paradox that using this knowledge of its surface and volume you could fill it with a bucket of paint but the same volume of paint would not be enough to paint its surface.

This paradox is resolved because the surface we generated has no thickness and you can’t find any real-life objects with no thickness which you could paint. Paint itself has finite thickness bounded by the radius of an atom.

Gabriel’s horn

Another thing you can think about is if you ever find real- life version of Gabriel’s horn or Gabriel’s trumpet could you play it?

Well no, since this trumpet is infinitely long, it will take you infinitely many years to come to its end, and even if you’re feeling especially adventurous and reach its end, it would be infinitely small so you couldn’t blow in it.