Define the concepts of integrable function and Riemann integral for functions of two variables (across a rectangle and over an arbitrary area).
I know how to define for a rectangle but not an arbitrary area.
EDIT:
Are the following correct?
In order for a function to be integrable, you have to function is limited and defined in a rectangle, and that for each real number e> 0 exists two staircase functions such that g (x, y) <= f (x, y) <= h (x, y) and Doubble integral of g (x, y) - Doubble integral of h (x, y) < e
The riemann integral of such a function is the double integral of funtionen f over the rectangle?
For an arbitrary area D, the function must be continuous in D and every (x, y) in D must be of the form p(x) <= y <= r(x), a <= x <= b.
D is compact => f is uniform continuous and bounded on D.
Brink (don't know if it's the right name in english) of D is a null set since p(x) and r(x) are continuous. Therefore, f is integrable over D ?
For an arbitrary area, you consider every possible way to get finitely many disjoint small rectangles that are contained in your rectangle, and compute the min value of the function in each rectangle and add up the weighted areas. The supremum over all choices of rectangles gives you a lower bound for your integral. Then you similarly compute the infimum over all choices of rectangles when you take the max value of the function within each rectangle, to get an upper bound for your integral. If the lower and upper bounds agree then the function is integrable over the area and the integral value is equal to the upper/lower bound.