Type I and Type II regions

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Type I and Type II regions
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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

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

The 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

$y$ :

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

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