Two cylinders of unit diameter intersect (100%) at right angles.

If you imagine that the cylinders are centered on the X and Y axes, and
look at the thing from the direction of the Z axis, you can see that
the cross section of this volume at any point on the Z axis is a
square.  The side of the square as you go from Z=-1/2 to +1/2 is equal
to 2*SQRT(1/4-Z^2).  So to get the volume you evaluate the integral of
(1 - 4Z^2) dZ from -1/2 to 1/2.  What does that come to?  Z - 4Z^3/3?
or something?  Obviously I have integrated maybe one equation in the
last 15 years.  If this were right by some odd chance then the volume
would work out to 2/3.

-- Pete Kimball <achepak@interaccess.com>

[ Editor: the integral is right: [Z - 4Z^3/3] from -1/2 to 1/2 is
(1/2 - 1/6) - (-1/2 + 1/6) = 2/3. ]

[ Editor: problem posed by Aubrey Jacobus.  See also problem 1 of
geometry/3d.problems. ]