Torricelli’s Trumpet (Gabriel’s Horn)
Torricelli’s trumpet is a shape that has an infinite surface area while maintaining a finite volume.
Taking the graph of f(x) = 1/x for all x >= 1 and revolving it about the x axis creates the shape shown above. Computing the volume of the shape from 1 to any point greater than 1, a, the following can be shown:
This shows that as a increases, the volume gets closer and closer to the value pi. Taking the limit of this solution as it approaches infinity, the value converges to pi–because a gets infinitely big, 1/a becomes 0.
Calculating the surface area using the equation, similar to above, from 1 to any point a, gives the following:
Unlike the solution to the volume equation, taking the limit of a as it approaches infinity for this solution diverges, because as aapproaches infinity, ln(a) approaches infinity. Therefore, with an infinite surface area, there is an finite volume.
*images of computations taken from wikipedia*