Our original concept of area was defined in terms of an integral over a line. We broke the line into small intervals and used these to divide the area under the curve defined by f into strips.
There is nothing to stop us, however, from breaking the whole area into small pieces and adding these up.
Thus if we have an area A we can put a grid of tiny squares over it and count how many squares are inside it, and how many on its boundary, and estimate its area from those numbers (and the size of the squares).
We can estimate volume under a surface defined by f(x, y) similarly by multiplying the area of each square by the value of f at a point inside it; again we can define Riemann sums and consider the integral defined if, as the maximum area of a piece goes to zero, all Riemann sums approach a specific value.
An integral of this kind is called an area integral, and we can denote it as
Again if f is continuous over a bounded closed region such integrals will always be well defined.