Көпөлшемді интегралдар: Жалданулар мен қасиеттері

Generalization of definite integrals to functions of multiple variables

In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Physical (natural philosophy) interpretation: S any surface, V any volume, etc Incl. variable to time, position, etc. Integrals of a function of two variables over a region in (the real number plane) are called double integrals, and integrals of a function of three variables over a region in (real number 3D space) are called triple integrals. For multiple integrals of a single variable function, see the Cauchy formula for repeated integration.

Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three dimensional Cartesian plane where ) and the plane which contains its domain. Since the concept of an antiderivative is only defined for functions of a single real variable, the usual definition of the indefinite integral does not immediately extend to the multiple integral.

Multiple integrals have many properties common to those of integrals of functions of one variable (linearity, commutativity, monotonicity, and so on). One important property of multiple integrals is that the value of an integral is independent of the order of integrands under certain conditions. This property is popularly known as Fubini's theorem.

If D is normal with respect to the y axis and f : D → 'R' is a continuous function; then α(y) and β(y) (both of which are defined on the interval ) are the two functions that determine D. Again, by Fubini's theorem:

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In case of unbounded domains or functions not bounded near the boundary of the domain, we have to introduce the double improper integral or the triple improper integral.

Triple integral was demonstrated by Fubini's theorem. Drichlet theorem and Liouville 's extension theorem on Triple integral.