Take a cookie dough rolled flat into a perfect circle of radius R, and wrap it around a cylinder of radius R/Pi , such that opposite points of the original circle now meet at the top. After the cookie is baked and hard, remove the cylinder and fill with cream cheese.

Scrape off the excess filling using a straight edge held perpendicular to the long axis and connecting symmetric points of the edges as you scrape.

What is the volume of one of these theoretical cannoli?

Very interesting finding by Joel, that the volume of this shape can be expressed in a way that involves a Bessel function.  The only other physical process (of which I was aware) that produces Bessel functions is holding one end of a rope that hangs down and then spinning the rope.  The shape of the rope, if I recall correctly, is a type of Bessel function.  There are probably others.

My own attack of this problem involved integrating with slices transverse to the longitudinal axis, the same orientation Eric used (integrating the other way).  I chose to break each slice into a fraction of a circle, plus a triangle.  I haven't yet attempted a solution of the integrals.

Posted by Larry on 2006-11-03 07:50:57