POINT 1. DOUBLE INTEGRAL
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
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-
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
. ( 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.
1 (linearity). For any functions and any constants
2 (additivity). If a domain is divided into two disjoint parts , (fig. 5), then