Properties of double integrals

Let $f(x,y)$ and $g(x,y)$ be continuous functions on a region $R$ then the following properties hold for double integrals:

$\displaystyle \iint_R c\, f(x,y)\,\dd(x,y)= c \iint_R f(x,y)\,\dd(x,y)$ for any constant $c$ .
$\displaystyle \iint_R \bigl(f(x,y)\pm g(x,y)\bigr)\,\dd(x,y)=\iint_R f(x,y)\,\dd(x,y) \pm \iint_R g(x,y)\,\dd(x,y)$ .
If $f(x,y)\ge 0$ on $R$ , then $\displaystyle \iint_R f(x,y)\,\dd(x,y)\ge 0$ and that double integral is equal to the volume under the graph on $R$ .
If $f(x,y)\le g(x,y)$ on $R$ , then $\displaystyle \iint_R f(x,y)\,\dd(x,y)\le \iint_R g(x,y)\,\dd(x,y)$ .
$\displaystyle \Biggl|\iint_R f(x,y)\,\dd(x,y)\Biggr|\le \iint_R \bigl|f(x,y)\bigr|\,\dd(x,y)$ . This is known as the triangle inequality.
If $R$ is the union of two non-overlapping subareas $R_1$ and $R_2$ , then $\displaystyle \iint_R f(x,y)\,\dd(x,y)= \iint_{R_1} f(x,y)\,\dd(x,y)+ \iint_{R_2} f(x,y)\,\dd(x,y)$ .
This is known as the additivity of regions of integration.
$\displaystyle \iint_R 1\, \dd(x,y) = \text{area of } R$ .

These properties sometimes come in handy when you change the order of integration in a double integral.