[Guest]

Is this possible? – Infinity

You have an object (plane figure) with one axis of symmetry (x), the area of this is infinite.

If you rotate around the x axis the object, you will get a body with finite volume.

Is it possible? (if it is, give one example, if not give an appropriate proof!)

I hope it's clear, if you need any further explanation, don’t hesitate to ask!

Edited March 11, 2010 by det

[Guest]

Is this possible? – Infinity

You have an object (plane figure) with one axis of symmetry (x), the area of this is infinite.

If you rotate around the x axis the object, you will get a body with finite volume.

Is it possible? (if it is, give one example, if not give an appropriate proof!)

I hope it's clear, if you need any further explanation, don’t hesitate to ask!

The shape could be the curve y^2 = 1/x^2 {x>1} This has symmetry about the x axis. Its area is: 2 * ln(infinity) = infinity.

If we take the volume of revolution: pi * integral(1,inf) (1/x^2) dx = pi.

Hope this is what your looking for.

[Guest]

Yep, nice solution psychic_mind!

[Guest]

i believe gabrial's horn, as posted by psychic mind fits the description.

i always found that interesting, you can't cover the surface with paint but you can fill it up.

[Guest]

i believe gabrial's horn, as posted by psychic mind fits the description.

i always found that interesting, you can't cover the surface with paint but you can fill it up.

Technically, you can never exactly fill the volume. Since PI has yet to be shown as an exact number, one can only under-fill or over-fill the volume. Unless you find a way to take a perfect sphere, or some other volumetric object, of volume PI and split it into infinitesimal parts in order to fill the aforementioned volume.

Edited March 11, 2010 by Egghead

[bonanova]

Technically, you can never exactly fill the volume. Since PI has yet to be shown as an exact number, one can only under-fill or over-fill the volume. Unless you find a way to take a perfect sphere, or some other volumetric object, of volume PI and split it into infinitesimal parts in order to fill the aforementioned volume.

Pi does not have an exact finite decimal representation.

But if you perform a change of scale, from integers to

circumferences per diameters, it has the very tractable

and precise value of unity.

Exactly filling the volume might remain impossible, tho,

as it could become difficult to count paint molecules

in these units.

[bonanova]

Any closed fractal has infinite length and encloses a finite area.

Some say the coastline of England has infinite length.

Rotating either of these will produce a finite volume enclosed by an infinite area.

[Guest]

Any closed fractal has infinite length and encloses a finite area.

Some say the coastline of England has infinite length.

Rotating either of these will produce a finite volume enclosed by an infinite area.

But in your solution the area of the 2D object is finite, so it does not meet the initial requirements; but it’s an interesting example finite area with infinite perimeter (y(x):=1/x^n n>1 xє[1,inf∞[ functions have the same property).

Edited March 12, 2010 by det

[Guest]

Technically, you can never exactly fill the volume. Since PI has yet to be shown as an exact number, one can only under-fill or over-fill the volume. Unless you find a way to take a perfect sphere, or some other volumetric object, of volume PI and split it into infinitesimal parts in order to fill the aforementioned volume.

Pi does not have an exact finite decimal representation.

But if you perform a change of scale, from integers to

circumferences per diameters, it has the very tractable

and precise value of unity.

Exactly filling the volume might remain impossible, tho,

as it could become difficult to count paint molecules

in these units.

Technically, you can never manufacture the bowl to fill

a) it requires infinite accuracy

b) the surface is infinite...

But, if we assume that we have the bowl and we are not limited by the size of the molecules (split it into infinitesimal parts), we can just pour the paint into it, if it’s too much it will overflow, so the volume can be integer, transcendental, etc. number…

Of course we have some other problems as well, like surface-tension. Moreover why do you want to fill technically and exactly that f* bowl? : )

Edited March 12, 2010 by det