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 _{1}∫^{a} (1/x^{2}) dx = π (1 – 1/a)

So, lim_{a->∞} π (1 – 1/a) = π (1 – 0) = π

The surface area can also be calculate as:

Surface Area = 2π _{1}∫^{a} √(1 + 1/x^{4}) / x dx

We know that √(1 + 1/x^{4}) > √(1)...so we know that the Surface Area > 2π _{1}∫^{a} √(1) / x dx = 2π ln a

So, lim_{a->∞} 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.

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## Pickett 13

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

:PIVolume= π_{1}∫^{a}(1/x^{2})dx= π (1 – 1/a)So, lim

_{a->∞}π (1 – 1/a) = π (1 – 0) =πThe surface area can also be calculate as:

Surface Area= 2π_{1}∫^{a}√(1 + 1/x^{4}) / xdxWe know that √(1 + 1/x

^{4}) > √(1)...so we know that the Surface Area > 2π_{1}∫^{a}√(1) / xdx= 2π ln aSo, lim

_{a->∞}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.

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