Double Integrals Over Rectangular Regions

    We defined the definite integral of a single variable as a limit involving the Riemann sums



where



are elements in a partition of the interval and is a representative in the subinterval We now apply the same idea to define a definite integral of two variables



over the rectangle

Definition (Riemann Sum) Partition the interval into subintervals and the interval into subintervals. Using these subdivisions, partition the rectangle into cells (subrectangles) and call this partition Choose a representative point from each cell in and form the sum,



where is the area of the -th representative cell. This is called the Riemann sum of with respect to the partition and the cell representation To measure the size of the rectangles in the partition we define the norm of the partition to be the length of the longest diagonal of any of the rectangle in the partition.

Definition (Double Integral) Refine a partition by subdividing the cells in such a way that the norm decreases. When this process is applied to the Riemann sum and the norm decreases to zero, we write



If this limit exists, its value is called the double integral of over the rectangle Indeed, if is defined on a closed, bounded rectangular region in the -plane, then the double integral of over is defined by  



provided this limit exists, in which case, is said to be integrable over

    It can be shown that if is continuous on a rectangle then it must be integrable on , although it is also true that certain discontinuous functions are integrable as well. Moreover, it can also be shown that if the limit that defines the integral exists, then it is unique in the sense that the same limiting value results no matter how the partitions and subintervals are chosen.

Proposition (Properties of Double Integrals) Assume that all the given integrals exist on a rectangular region

(i)  Linearity Rule: For constants and
    

    
(ii) Dominance Rule:  If throughout a rectangular region then
    


(iii) Subdivision Rule:  If the rectangular region of integration is subdivided into two (disjoint) subrectangles and then

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Cite this as:
Double Integral Over A Rectanglular Region
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/double-integral-over-a-rectanglular-region.html