Seems rather paradoxical in my opinion.
In: 4
As it turns out, dealing with infinite are both hard and non-intuitive.
First, lets simplify the problem a bit. The horn is a rotational integral of a curve like 1/x, so lets just work with 1/x instead. It has all of the same properties that matter for this paradox, so it will work.
What does 1/x do as it goes to infinity? It *converges* toward 0. While it gets extremely close to 0 and limit of 1/x as x approaches infinity is 0, it never actually reaches zero. This is a case where the difference actually matters.
Since 1/x is approaching 0, the distance between the x axis and the curve become infinitely small. This means that when you take the integral and add up all of those distances, the values near infinite don’t actually change the total very much which means it also converges towards a specific number as well. This means that we can bound the value of the integral between a pair of two other finite values, so it must be finite as well.
However, because 1/x never actually reaches 0, it must have infinite length (or surface areas as a 3d curve). By convention, a curve is “closed” and ends when it crosses the x axis. This means as the curve goes to infinity, that small distance between the curve and zero prevents it from closing and means that the length of the curve keeps increasing and doesn’t have an upper bond. This means that it must be infinite as well.
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