Type I and Type II regions

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

1. Chop $R$ into a bunch of small boxes.
2. Compute $f(x^*,y^*)\Delta x \Delta y$ for each box contained in $R$.
3. Add up the boxes, and
4. Take a limit as we chop $R$ into smaller and smaller pieces.

The exact same ideas apply when $R$ is an arbitrary region. As with rectangles, things get a lot simpler if we arrange the boxes intelligently.

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