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 Integrating the other way by Eric)

The following program evaluates your integral (of -sin(pi*sqrt(1-y^2))cos(pi*sqrt(1-y^2)) from y=0 to 1) numerically:

DECLARE FUNCTION acos# (x#)
DECLARE FUNCTION asin# (x#)
' here R is assumed = pi
DEFDBL A-Z
DIM SHARED pi
pi = 4 * ATN(1)
stp = .0000001
FOR y = 0 TO 1 STEP stp
 't = t - SIN(pi * SQR(1 - y * y)) * COS(pi * SQR(1 - y * y)) * stp
 t = t - SIN(2 * pi * SQR(1 - y * y)) / 2 * stp
NEXT
PRINT t

Note one version of the formula being integrated is commented out. The second one evaluates to the same as the formula is for half the sine of twice the angle in question.

The result comes out to about .16680485. So I take it that this determination comes to r^3 * .66680485, somewhat higher than what my result was.

Posted by Charlie on 2006-11-01 20:07:53