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## Question

So, I learned this one when I was in my Calculus class and thought it was just sort of fun. Basically, there is a shape, called Gabriel's Horn (where you take the curve generated by the function f(x) = 1/x (where x >= 1)) and revolve it around the x-axis to form a horn shape.

Using calculus, you can calculate the volume of this shape (it doesn't show up very well, but π is PI:

Volume = π 1a (1/x2) dx = π (1 – 1/a)

So, lima->∞ π (1 – 1/a) = π (1 – 0) = π

The surface area can also be calculate as:

Surface Area = 2π 1a √(1 + 1/x4) / x dx

We know that √(1 + 1/x4) > √(1)...so we know that the Surface Area > 2π 1a √(1) / x dx = 2π ln a

So, lima->∞ 2π ln a =

You might ask why this is a paradox...you can think of it this way:

Imagine you are a painter, it would take infinite amount of paint to paint the surface of this figure...however, you could FILL the entire shape with π volume of paint.

I like this one (I know WHY it isn't a paradox...but that's no fun to give that away right away...), so I thought I'd post it.

## Recommended Posts

• 0 Eh... I'm weak at calculus, but your analogy is interesting.

Something I don't understand...

What is a? Why it can become an infinite? Can u really use limit like that? What does it mean if a is infinite? Does the horn get bigger or what?

Edited by folg3nd

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• 0 Ah, math. I HATE math. everyone tells me it's usefull. How is it Usefull?

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• 0 Ah, math. I HATE math. everyone tells me it's usefull. How is it Usefull? Think about it SRP... How might you manage your retirement if you do not understand the math that involves your money? Nuff said there..

The Paradox of Gabriels Horn is about what can happen in its construction. It isn't about the exterior of the horn, as much as it is about painting the total interior..

Hang in there... Great proposal, Pickett

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