The difference between double integral and quadratic integral

The double integral is the integral about the area, and the second integral is the unary quadratic integral.

① When f(x, y) is continuous in a bounded closed region, the double integral and the quadratic integral are equal. This relationship does not apply to open areas or unbounded areas.

② Quadratic integration does not necessarily lead to double integration. For example, for [0, 1]*[0, 1] region, for any x∈[0, 1], a function g (x, y) (y ∈ [0,/kloc-] continuous to y can be defined.

(3) Double integration is not always possible. Region S={(x, y) | x > = 1，| y | & lt= 1/x^3}。 The identity function f (x, y) = 1, (x, y) ∈ s. F can be double-integrated on s, but not double-integrated (integrating x first, then integrating y, and integrating infinity on a straight line with y=0).

Integral commutation

In the example of ③ above, the integrals are reversed, one can be integrated and the other can't be integrated (first, the y integral x is fixed, and the integral will get 2/x 3.2/x 3 to x(x belongs to [1, infinity)).

Where x and y can be exchanged.

Continuous and absolutely integrable, the step-by-step integral of x or y exists. Under special circumstances, the function can switch X and Y continuously in the bounded closed region. At this time, because the function is continuous, there is an extreme value in the closed area.

Integral transformation must require that the transformed integral interval is the same as the original one, and there can be no repeated integral.