Properties of The Double Integral

We are now going to look at some properties of the double integral.

Theorem 1: Let $z = f(x, y)$ be a two variable real-valued function that is integrable over $D \subseteq D(f)$. Then: a) If $D$ has zero area, then $\iint_D f(x, y) \: dA = 0$. b) If $D$ has area $d$, then $\iint_D f(x, y) \: k = kd$. c) $\iint_D \left ( f(x, y) + g(x, y) \right ) \: dA = \iint_D f(x,y) \: dA + \iint_D g(x,y) \: dA$ (Addition Property). d) $\iint_D kf(x, y) \: dA = k \iint_D f(x,y) \: dA$ (Scalar Multiple Property). e) If $f(x,y) ≤ g(x,y)$ for all $(x, y) \in D$ then $\iint_D f(x,y) \: dA ≤ \iint_D g(x,y) \: dA$. f) $\biggr \rvert \iint_D f(x,y) \: dA \biggr \rvert ≤ \iint_D \mid f(x,y) \mid \: dA$. g) If $D_1, D_2, ..., D_n \subseteq D(f)$ are non-overlapping subsets of $D(f)$ that share no interior points with each other and $D = \bigcup_{k=1}^{n} D_k$ then $\iint_D f(x,y) \: dA = \sum_{k=1}^{n} \iint_{D_k} f(x,y) \: dA$ (Additivity of Domains Property).