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 $x$:$$\iint_R f(x,y)\, dA = \int_{x=a}^b\, \int_{y=g(x)}^{h(x)}
\,f(x,y)\, dy \, dx$$
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
$y$:$$\iint_R f(x,y)\, dA = \int_{y=c}^d \,\int_{x=g(y)}^{h(y)}
\,f(x,y)\, dx\, dy$$
