What is the VOLUME of the union of these two cylinders:

x^2 + z^2 = 4
y^2 + z^2 = 4

You can divide the region into 8 octants, solve for one octant volume,
and then multiply by 8.  It would probably be best to use a double
integral.

This one you can do without calculus, if you're clever about it. 
Every cross section of the solid of intersection perpendicular to the
z axis is a square.  Every such square fits tightly around a sphere of
radius 2, centered at the origin.  So the ratio of the volume of the
solid of interesection to the volume of the sphere is the same as the
ratio of areas of a square and the circle that just fits inside: 4/pi.
 The volume of a sphere of radius 2 is 32 pi/3, so the volume of your
solid is 128/3.

Although the previous comment is more elegant, a purely calculus
approach can be done if one does not rush into polar coordinates - as
I did when I first tried to do it. Consider only the first octant and
multiply by 8. Think of the base of the figure to be a quarter circle
in the xz plane and the height to be y=sqrt(4 - z^2). Form the double
integral with dx inside dz. The limits on the outside integral are 0
and 2. The limits on the inside integral are 0 and sqrt(4 - z^2). This
makes the inside integral trivial. The second integration is just of
(4 - z^2). Again the answer comes out 128/3.

Both approaches are elegant in my book, but I definitely did not think
of the first one! I obtained 128/3 when I did it myself using
Mathematica, but I just wanted to confirm.  Thanks a lot for the help
you two.