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Torricelli's trumpet
Torricelli's trumpet (also Gabriel's horn) has a finite volume and an infinite surface.
Explanation
We start from the function and use the graph, in which we avoid dividing by zero.
We rotate the function around the x-axis and calculate the volume and the surface between x = 1 and x = a where a > 1.
The drawing gives an impression of the trumpet.
Volume
The formula for the volume is
so applies
Surface
The formula for the surface area is
so applies
Paradox
It seems like a paradox, that the volume of this trumpet has a real value but the surface is infinitely large. With mathematics, you can calculate things which we cannot comprehend intuitively. The picture is just a tool. The trumpet does not end with a mouthpiece but always continues. The calculation shows that everywhere a surface exists but no volume is added anymore
HistoryThis figure was studied by Italian physicist Evangelista Torricelli in the 17th century. |