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Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 51', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 318 Integrals 7.3.1-2. Classes of integrable functions. Further on it is assumed that D is a closed bounded domain. 1. If f x y is continuous in D then the double integral f x y dx dy exists. 2. If f x y is bounded and the set of points of discontinuity of f x y has a zero area e.g. the points of discontinuity lie on finitely many continuous curves in the x y plane then the double integral of f x y over the domain D exists. 7.3.1-3. Properties of the double integral. 1. Linearity. If functions f x y and g x y are integrable in D then yy af x y bg x y dxdy a f x y dxdy b g x y dxdy where a and b are any numbers. 2. Additivity. If the domain D is split into two subdomains D1 and D2 that do not have common internal points and if the function f x y is integrable in either subdomain then f x y dxdy f x y dxdy f x y dx dy. J J D J JD1 J jd2 3. Estimation theorem. If m f x y M in D then mS jj f x y dxdy MS where S is the area of the domain D. 4. Mean value theorem. If f x y is continuous in D then there exists at least one internal point x y g D such that 4 f x y dx dy f y s . The number f x y is called the mean value of the function f x y in D. 5. Integration of inequalities. If x y f x y g x y in D then yy x y dxdy f x y dxdy g x y dx dy. In particular if f x y 0 in D then f x y dx dy 0. 6. Absolute value theorem d f x y dxdy D f x y dxdy. 7.3. Double and Triple Integrals 319 Figure 7.4. A double integral of a nonnegative function f x y over a domain D is equal to the volume of a cylindrical body with base D in the plane z 0 and bounded from above by the surface z f x y . 7.3.1-4. Geometric meaning of the double integral. Let a function f x y be nonnegative in D. Then the double integral f x y dx dy is equal to the volume of a cylindrical body with base D in the plane z 0 and bounded from above by the surface z f x y see Fig. 7.4. 7.3.2. Computation of the Double Integral 7.3.2-1. Use of iterated integrals. 1 . If a domain D is defined in the x y plane by the .
