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Torricelli's trumpet

Torricelli's trumpet (also Gabriel's horn) has a finite volume and an infinite surface.



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.



The formula for the volume is

so applies



The formula for the surface area is

so applies



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



This figure was studied by Italian physicist Evangelista Torricelli in the 17th century.

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