The surface area of surface S defined on a rectangular region R can be defined as
where 
is the small patch of surface area of the tangent plane
corresponding to the 
sub-rectangle of a rectangular region.
The area of the surface with equation z=f(x,y), 
, where 
and 
are continuous, is
or
The surface area of a surface can be estimated as a sum of patches; as the patches get smaller and smaller, the estimate gets better and better. In figure 2, p. 1040, the author carefully derives the surface area of a planer region overlying a small rectangular region. If you understand this derivation, you understand the formula!