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?

(In reply to solution--but there's plenty of room for error by Charlie)

I used a different though I believe similar integral (integrating over the angle from the centor of the cylinder and arrived at the same result from Mathematica :

about 0.533802 r³

However, I was able to get a somewhat symbolic solution to the integral:

r³(Pi-BesselJ[1,2Pi])/2Pi

Where BesselJ is a Bessel Functon of the first kind, so BesslJ[1,2Pi] =  J sub 1(2Pi)

I got this via:

4*(r/Pi)³*Integral from 0 to 2Pi of (sin²x)*sqrt(Pi²-x²)dx

Posted by Joel on 2006-11-02 21:57:08