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Double integrals over a rectangle.
Let f(x,y) be a function defined on a rectangular domain
.
We divide the rectangle Rwith a network of lines parallel to the coordinate axes, and we number
the little rectangles A_{1}, A_{2}, and so on. On each little
rectangle A_{k} we choose a point
(x_{k}, y_{k}), as on Fig 1. The area of such a little rectangle is denoted
.
Figure 1:
Partitioning of a rectangular region.

Define
.
If this sum
has a finite limit when we refine indefinitely the network, then this
limit is called the double integral of f over R and is
denoted
Proposition 7.1
 1.

.
 2.

.
 3.
 If
,
then
.
 4.
 If
,
then
.
Proposition 7.2
Suppose that the rectangle
R is the union of two rectangles
R_{1} and
R_{2}, as dispayed on Fig.
2, then:
Figure 2:
The union of two rectangles.

Such a double integral can be viewed as the volume of the threedimensional region enclosed by the domain R in the xyplane, the graph of f and the planes parallel to the coordinates axes and perpendicular to the sides of R.
Theorem 7.3 (Fubini)
If
f is continuous in a region containing the rectangle
R, then:
Example 7.4
f(
x,
y)=
xy+1 over
.
Next: Double integrals over a
Up: Double integrals.
Previous: Double integrals.
Noah DanaPicard
20010530