Double Integral: Definition and Properties

If $f(x,y)$ is continuous throughout the rectangle region $R:x\in [a,b], y\in [c,d]$, then

$$\begin{aligned}&\iint_R f(x,y) \mathop{dA} \\ =&\int_a^b\int_c^d f(x,y) \mathop{dy}\mathop{dx} \\ =&\int_c^d\int_a^b f(x,y) \mathop{dx}\mathop{dy}\end{aligned}$$

Let $f(x,y)$ be continuous over the region $R$

• If $R$ is defined by $x\in [a,b], y\in [g_1(x), g_2(x)]$ with $g_1,g_2$ continuous on $[a,b]$, then

$$\iint_R f(x,y) \mathop{dA}=\int_a^b\int_{g_1(x)}^{g_2(x)} f(x,y) \mathop{dy}\mathop{dx}$$

• If $R$ is defined by $y\in [c,d], x\in [h_1(y), h_2(y)]$ with $h_1,h_2$ continuous on $[c,d]$, then

$$\iint_R f(x,y) \mathop{dA}=\int_c^d\int_{h_1(y)}^{h_2(y)} f(x,y) \mathop{dx}\mathop{dy}$$