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On the practical use of sciencesΒ Β
There are tons of functions with similar properties, consider one ...
simple hyperbola Y = 1 / X
The area under the curve is infinite.
Even if we try to "cut out" a part - a figure of finite height between the curve Y=1/X and the 0X axis, from X=1 to infinite XΒ
(the colored piece on Fig.1) ...
It is impossible: an infinite area (under curve) requires infinite amount of paper? π
Mathematics warns against wrong attempts:
Fig.1.
Area under curve =
On the other hand, the volume of the solid of revolution (the "funnel" formed by the rotation of a semi-infinite piece of hyperbola around the 0X axis - see Fig. 2) is a finite value.Β See below:
Let's calculate that volume by summing up the volumes of the narrow cylinders with a base of variable radius on common 0X axis.
Cylinder base area is:
Fig.2.
in our case:Β
So, the volume of the funnel (from x = 1 to infinity) is:Β
Interesting fact.
But suddenly ... a clever man appears and said:
- Let's do a thought experiment: take the required FINITE volume of paint, fill a funnel (Fig. 2) with it. Then let's dip a flat shape (Fig. 1) into the funnel ... and cover the INFINITE area with a FINITE amount of paint ...
And the fairy tale will come true?Β
Confusion !!!Β
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