Double integrals over a rectangular region

Notes:

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

Double Integrals Over a Rectangular Region

If $f(x,y)$ is continuous on a rectangular region $R=[a,b] \times [c,d]$, then

$\int \int _R f(x,y) dA$

gives the volume under the function within that region. The integral is also represented as:

$\int \int_R f(x,y) dA = \int_a^b \int_c^d f(x,y)dydx$
$= \int _c^d \int_a^b f(x,y)dxdy$

These integrals are known as iterated integrals.

Double Integrals of Products

If $f(x,y)=g(x)h(y)$ is continuous on a rectangular region $R=[a,b] \times [c,d]$, then

$\int \int_R f(x,y)dA = \int \int_R g(x)h(y)dA$

which can be rewritten as:

$\int^b_a \int^d_c g(x)h(y)dydx = (\int^b_a g(x)dx ) ( \int^d_c h(y)dy)$