How it works ?
Let’s explore GABRIEL’S HORN: GABRIEL’S HORN = one bizarre paradox! This surface is formed by rotating the graph of the function f(x)=1/x about the X-AXIS for 1<= ∝ <= ∞ (right branch of this hyperbola).
If you evaluate the improper integral that gives the volume of such a solid of revolution, you get a finite value. Yet if you evaluate the improper integral that yields its surface area, you’ll find that it diverges (i.e. is infinite).
Thus, we have an infinitely long trumpet we can fill with paint (i.e. has finite volume), but for which we cannot have enough paint to paint its surface (i.e. has infinite surface area).
How can we make sense of this phenomenon ???
GABRIEL’S HORN: Note that the function described above has been shifted LEFT 6 UNITS.