The paradox
known as "Gabriel's
Horn" deals with the
solid formed when the
graph y=
1/x is
rotated around the x-axis, considered when x≥1. It
looks a bit like a horn (hence the name) with its wide
opening at x=1, facing the origin, and
narrowing toward the
x-axis as x→
∞.
The volume of any solid formed by rotation of a graph over a≤x≤b with y=r around the x-axis is:
b
∫ πr2dx
a
Therefore, volume of the horn solid is the
improper integral
∞ ∞
π ∫ (1/x)2dx=-π1/x ] = π
1 0
It turns out that the volume is exactly equal to π.
The surface area of any solid formed by rotation of a graph over a≤x≤b with y=r around the x-axis is:
b
∫ 2πr√(1+(dy)2)dx
a
The surface area of this solid is another
improper integral:
∞
∫ 2π1/x√(1+(d/dx 1/x)2)dx=∞
1
The
proof of this is a bit on the
lengthy side, but you can
trust me that it's
true. Or, if you have a
worthwhile calculator like the
TI-89, you can just type it in and prove it to
yourself.
So, this solid has a finite volume (V=π), but an infinite surface area! This means that if the horn was a paint can, it would not be able to hold enough paint to cover its own surface. However, this is seemingly impossible; think of it this way. Since the can would be full, the inside surface of the can (which, of course is the same as the outside surface, since the thickness of the can is 0) would be covered by paint. However, this area is so large that no amount of paint could possible cover it. Somehow, our π units of paint are covering an infinite surface area, and still filling in the space between the walls. Therefore, π>∞??
There's a good 3D picture of the Horn, plus a humorous one of the "Paint-Can Paradox" at "http://www.geocities.com/Eureka/Plaza/4033/gabriel_index.en.html". You can play with an excellent 3D model of the Horn at "http://mathworld.wolfram.com/GabrielsHorn.html".