theorem: properties of double integrals

Assume that the functions [latex]f(x,y)[/latex] and [latex]g(x,y)[/latex] are integrable over the rectangular region [latex]R[/latex]; [latex]S[/latex] and [latex]T[/latex] are subregions of [latex]R[/latex]; and assume that [latex]m[/latex] and [latex]M[/latex] are real numbers.

1. The sum [latex]f(x,y) + g(x,y)[/latex] is integrable and:
   [latex]\underset{R}{\displaystyle\iint}[f(x,y)+g(x,y)]dA=\underset{R}{\displaystyle\iint}f(x,y)dA+\underset{R}{\displaystyle\iint}g(x,y)dA[/latex]
2. If [latex]c[/latex] is a constant, then [latex]cf(x,y)[/latex] is integrable and:
   [latex]\underset{R}{\displaystyle\iint}cf(x,y)dA = c\underset{R}{\displaystyle\iint}f(x,y)dA[/latex]
3. If [latex]R = S \cup T[/latex] and [latex]S \cap T = \emptyset[/latex] except an overlap on the boundaries, then:
   [latex]\underset{R}{\displaystyle\iint}f(x,y)dA = \underset{S}{\displaystyle\iint}f(x,y)dA + \underset{T}{\displaystyle\iint}f(x,y)dA[/latex]
4. If [latex]f(x,y)\geq g(x,y)[/latex] for [latex](x,y)[/latex] in [latex]R[/latex], then:
   [latex]\underset{R}{\displaystyle\iint}f(x,y)dA\geq \underset{R}{\displaystyle\iint}g(x,y)dA[/latex]
5. If [latex]m \leq f(x,y)\leq M[/latex], then:
   [latex]m \times A(R) \leq \underset{R}{\displaystyle\iint}f(x,y)dA \leq M \times A(R)[/latex]
6. In the case where [latex]f(x,y)[/latex] can be factored as a product of a function [latex]g(x)[/latex] of [latex]x[/latex] only and a function [latex]h(y)[/latex] of [latex]y[/latex] only, then over the region [latex]R = { (x,y)\mid a \leq x \leq b,c \leq y \leq d }[/latex], the double integral can be written as:
   [latex]\underset{R}{\displaystyle\iint}f(x,y)dA = \left(\displaystyle\int_a^bg(x)dx\right)\left(\displaystyle\int_c^dh(y)dy\right)[/latex]