If $R$ is a rectangle
in the $x$-$y$ plane and $f(x,y)$ is a function defined on $R$
then we saw that $\displaystyle\iint_R f(x,y)\, dA$ is what we get
when we

Chop $R$ into a bunch of small boxes.

Compute $f\left(x^*,y^*\right)\,\Delta x\, \Delta y$
for each box contained in $R$.

Add up the boxes, and

Take a limit as we chop $R$ into smaller and smaller
pieces.

The same ideas apply when $R$ is an
arbitrary region that is not a rectangle. As
with rectangles, things get a lot simpler if we arrange the boxes
intelligently. This video explains the concepts.

The video above explained why we get the following summary of
integrals over an arbitrary region. Remember, $g$ and $h$ are functions on the $xy$-plane,
while $f(x,y)$ is a surface over (or
under) the $xy$-plane.

Type I regions
are regions that are bounded by vertical lines $x=a$ and
$x=b$, and curves $y=g(x)$ and $y=h(x)$, where we assume
that $g(x) < h(x)$ and $a < b$. Then we can integrate
first over $y$ and then over $x$:$$\iint_R f(x,y)\, dA =
\int_{x=a}^b\, \int_{y=g(x)}^{h(x)} \,f(x,y)\, dy \, dx$$

instruct.math.lsa.umich.edu

Type II regions
are bounded by horizontal lines $y=c$ and $y=d$, and curves
$x=g(y)$ and $x=h(y)$, where we assume that $g(y)< h(y)$
and $c < d$. Then we can integrate first over $x$ and
then over $y$:$$\iint_R f(x,y)\, dA = \int_{y=c}^d
\,\int_{x=g(y)}^{h(y)} \,f(x,y)\, dx\, dy$$