ABSTRACT
The (double) Riemann sum of f (x; y) corresponding to the specic partition P and the specic choice of points (i; j) 2 Rij is S (f) =
Pm j=1 f (i; j)
xiyj . If there exists a nite limit of the Riemann sums of f (x; y), as approaches 0, which does not depend on a choice of partitions and points (i; j), then this limit is called the double (Riemann) integral of f (x; y) over R, is denoted as follows: lim
!0 S (f) =
RR R fdA or lim
!0 S (f) =
RR R fdxdy, and
the function f (x; y) is called (Riemann) integrable over R. 2. Integrability over a general set. Let f (x; y) be dened on a bounded
of Analysis
closed set D R2 whose boundary @D has zero area. (The last means that for any " > 0 the boundary @D can be covered by a set of rectangles with the total area less than ". Those familiar with the measure theory will immediately recognize the denition of the plane Lebesgue measure zero. In particular, any rectiable curve has zero area.) Consider an arbitrary rectangle R = [a; b] [c; d] containing the set D and dene the following function on R: F (x; y) =
f (x; y) ; (x; y) 2 D 0; otherwise
. If F (x; y) is integrable over R, then f (x; y)
is integrable over D and RR
D fdA =
RR R FdA.
