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.