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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 A1, A2, and so on. On each little rectangle Ak we choose a point (xk, yk), as on Fig  1. The area of such a little rectangle is denoted .

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 R1 and R2, as dispayed on Fig.  2, then:

Such a double integral can be viewed as the volume of the three-dimensional region enclosed by the domain R in the xy-plane, 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 Dana-Picard
2001-05-30