University Mathematics Handbook (2015)
VIII. Integral Calculus of Multivariable Functions
Chapter 2. Double Integral
2.1 Definition of Double Integral
Let 
be a function defined above region 
. Let region be divided by lines parallel to coordinate lines, into elementary sections, resulting in a rectangular net covering . The area of inner rectangles 
is 
. Inside every rectangle, we choose a point 
and express a sum by all inner rectangles:
This is the integral sum of function corresponding to the given partition and dependent on the choice of points 
.
a. Function is integrable (according to Riemann) above domain if there exists a finite limit of its integral sum when the maximum diagonal of rectangles tends to zero, when the limit is independent both of manner of partitioning and of the choice of points .
This limit is the double integral of function over . It is denoted 
or 
.
b. Let 
be a curve in and 
be rectangles covering its area. is called zero area curve if 
.
c. Function is piecewise continuous in if it discontinuous at no more than a finite number of zero area curves.
d. If is continuous over then it is integrable.
e. If is integrable over it is bounded in .
f. If is bounded and piecewise continuous in , then it is integrable.
2.2 Properties of Double Integral
a. The area of domain : 
.
b. If functions and 
are integrable in , then:
1. For every real 
and 
, function 
is integrable, and
2. Function 
is integrable over .
3. If, in addition, 
on every point of , then:
c. If is integrable over , then 
is integrable in the same domain, and
The inverse is incorrect. That is, from the integrability of 
does not necessarily follow the integrability of in the same domain .
d. If is integrable in , 
, 
, then 
, when 
is the area of region .
If, in addition, is connected and is continuous in , then there exists a point 
in such that 
.
f. If function is integrable over , and region is divided, by curve of zero area, into connected regions 
and 
with no inner common points, then is integrable over and and holds:
