Gabriels Horn

Gabriels horn is the 3d object generated by a solid rotation of a plot of 1/x.

As the Wikipedia article says, it has some interesting properties, including a paradox indicating a finite volume, but infinite surface area

Beginning with a plot of 1/x,

we rotate about the x-axis by $2pi$ ,

The volume of the horn can be obtained by treating the horn as a stack of infinitesimal discs. Each disc has the area $pir^2$ , where r is a function of x. Evidently, this radius is the value of y for a plot of 1/x. Thus,

$a=pir(x)^2$ , where $r(x)=1/x$

The volume can be obtained by integrating the disc area,

$V=int a(x)dx$

Hence the volume,

$V=pi int_1^(x) (1/x)^2dx = pi [-1/x]_1^(x)$

Insert the limits we obtain,

$V=pi (1 - 1/x)$

Similarly, the surface area can be calculated by adding the circumferences of a stack of infinitesimal discs. Each disc has a circumference $c=2pir$ , where, as before, $r(x)=1/x$ . Thus,

$A=2pi int_1^x (dx)/x=2pi[lnx]_1^x=2pi[(ln1+lnx)=2pilnx$