Gabriel's Horn

Gabriel's Horn ( also Torricelli's trumpet) is discovered by Evangelista Torricelli body having an infinite area, but finite volume. The name derives, first, from the wind instrument resembling the form, on the other hand from the tradition here to view the Archangel Gabriel as the angel who blows the horn to announce Judgment Day, in which case the infinity is associated with the divinity.

Mathematical definition

Gabriel horn arises when the graph of the function of the domain ( the pole at x = 0 to be avoided) can rotate about the x-axis. Volume () and area ( ) of this rotating body are calculated in the following way:

To understand this, one must know that for all there with the improper integral from 1 to, ie to a finite value converges:

In contrast, the area under the curve is unbounded, because a primitive function of this curve is the natural logarithm:

Explanation of the paradox

On one hand, one would need an infinite amount color to completely cover the inner or outer surface of the horn, on the other hand, one could fill the inside of the horn with a finite amount of paint. The resolution of this apparent paradox is that for coating the entire inner surface of the body, a color layer having a small thickness as would be necessary for this have virtually no volume.