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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

 


Additional information

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

 


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