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Def. 1.Let a function of two variables be given in a some domain D of the -plane (fig. 1).

1. We divide the domain into n parts with areas and diameters .

2. We take arbitrary point in every part , find the value of the function at this point and multiply it by the area of .

3. We add all these products
Fig. 1 and get an integral sum


4. Let and . If there exists the limit of the integral sum , then this limit is called the double integral of the function over the domain D and is denoted by

( 1 )

Theorem 1(existence of a double integral).If a function is continuous in a domain D then its double integral over D exists.

It is evident that for a double integral gives the area of the domain D,

. ( 2 )

Mechanic sense of a double integral. If is the surface density of a plate , then its mass equals the next double integral

( 3 )

Anelement of the mass


it is the mass of the element with the area and with a constant surface density (fig. 2). Sum ofallthese elementsgives the mass of the plate which is re-
Fig. 2 presented by a double integral (3).

Def. 2.A cylindrical body [a curvilinear cylinder] is called a body bounded:

a) above by a surface ;

b) below by a domain D;

c) aside by a cylindrical surface with the generatrix parallel to -axis (fig. 3).

Geometric senseof a double integral. The volume of a cylindrical body equals the double
Fig. 3 Fig. 4 integral

. ( 4 )

■An element of the volume

is the volume of a right circular cylinder with the base of the area and the height (fig. 4). The volume of a cylindrical body is the sum of all these elements and is represented by the double integral (4).■

Properties of a double integral are the same as for a definite integral.

For example:

1 (linearity). For any functions and any constants


2 (additivity). If a domain is divided into two disjoint parts , (fig. 5), then

Fig. 5


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