Gabriel’s Horn — an oldie but still a goodie!

Here is a “paradox” that one encounters in second semester Calculus. Start with the function y = 1/x in the first quadrant, which is a connected piece of the usual rectangular hyperbola. Consider the curve in the range x ≥ 1. Rotate this curve about the X-axis to generate a 2 dimensional “horn” or infinite “bucket” in 3 dimensional space i..e, something like this:

torricelli

Now using standard Calculus 2 techniques one may compute separately the volume of this object (how much stuff it can hold) and its surface area (how large an area to paint). And the calculation looks this:

Equations                         Equations

Hmm! You see the problem — an object with finite volume but infinite surface area!! So if you were to fill it with paint there is only so much paint it can hold. But if you were to paint it, then no amount of paint would be enough. Can you explain the paradox??

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One thought on “Gabriel’s Horn — an oldie but still a goodie!

  1. Why is the thickness of the layer of paint on the far end of the horn considered to be finitely thick while it covers an infinitely small volume?
    It is equivalent to painting a infinitely thin line with a finitely thick layer of paint. How is it a paradox?

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