Double integrals over a general region

Double integrals of a multi-variable function give the volume under the function $f(x,y)$

Suppose the region is not rectangular, but rather the region is between two curves. For example, we want to integrate within the following region $D$, where $D$ is:

$a \leq x \leq b$
$g_1 (x) \leq y \leq g_2(x)$

Then the iterated integral will be:

$\int \int_D f(x,y)dA = \int_a^b \int_{g_1(x)}^{g_2(x)}f(x,y)dydx$

Likewise, suppose we have the region $D$, where $D$ is:

$g_1 (y) \leq x \leq g_2(y)$
$a \leq y \leq b$

Then the iterated integral will be:

$\int \int_D f(x,y)dA = \int_a^b \int_{g_1(y)}^{g_2(y)} f(x,y)dxdy$

You usually find the region $D$ yourself.

The three properties of double integrals are the following:

$\int \int_D f(x,y) + g(x,y) dA = \int\int_D f(x,y)dA \int \int_D g(x,y) dA$
$\int \int_D cf(x,y) dA = c \int \int_D f(x,y) dA, \mathrm{where\;c\; is\;a\;constant}$
$\int \int_D f(x,y) = \int \int_{D_1} f(x,y) + \int \int_{D_2} f(x,y) \;\mathrm{where\; D\; is\; split \; into\; 2\; regions\; D_1 \; \& \; D_2}$

Suppose you want to find the volume of a region that is above $g(x,y)$ and below $f(x,y)$, bounded by a region $D$. Then, the volume is:

$\int \int_D f(x,y) - g(x,y)dA$