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5.11 Properties of double integrals.

For a region $R$ , functions $f$ and $g$ , and a constant $C$ , the following hold:

\int\!\!\!\int_{R}(f+g)\,dxdy=\int\!\!\!\int_{R}f\,dxdy+\int\!\!\!\int_{R}g\,dxdy;

\qquad(i)

\int\!\!\!\int_{R}Cf\,dxdy=C\int\!\!\!\int_{R}f\,dxdy.

\qquad(ii)

$(iii)$ For disjoint regions $S$ and $T$ with union $R=S\cup T$ :

\int\!\!\!\int_{R}f\,dxdy=\int\!\!\!\int_{S}f\,dxdy+\int\!\!\!\int_{T}f\,dxdy.

$(iv)$ The area of the region $R$ is $A=\int\!\!\!\int_{R}dxdy.$

$(v)$ If $m\leq f(x,y)\leq M$ for all $(x,y)$ in $R$ and $R$ has area $A$ , then

mA\leq\int\!\!\!\int_{R}f(x,y)\,dxdy\leq MA.