The surface integral of some function f(x, y, z) over the
graph of a function g(x, y) is illustrated here. One window
shows the "shingles" of g(x, y), colored according to the
value of f(x, y, z),
and the second displays the value of the integral as the volume above
the x-y plane minus the volume below the x-y plane. Note that each
rectangular prism in this second window has a base equivalent in area
to that of a corresponding shingle in the first window, and a height
corresponding to the value of f(x, y, z) for that shingle.