PX153 - I3 - domain of integration

the region, $R$ , is divided into $m \times n$ rectangles by dividing the interval $a \leq x \leq b$ into $m$ stripes, and $c \leq y \leq d$ into $n$ stripes
each of the rectangles has area, $Δ A_{r s} = (x_{r} - x_{r - 1}) (y_{s} - y_{s - 1}) = Δ x_{r} Δ y_{s}$
the volume of the column above $Δ A_{r s}$ is:

f ({\bar{x}}_{r}, {\bar{y}}_{s}) Δ x_{r} Δ y_{r}

where, $({\bar{x}}_{r}, {\bar{y}}_{r})$ is some point in the range $x_{r - 1} \leq {\bar{x}}_{r} \leq x_{r}$ and $y_{s - 1} \leq {\bar{y}}_{s} \leq y_{s}$

the total volume is then:

I = \sum_{s = 1}^{m} \sum_{r = 1}^{n} f ({\bar{x}}_{r}, {\bar{y}}_{s}) Δ x_{r} Δ y_{r}

taking the limit $n \to \infty$ :

I = \sum_{s = 1}^{m} (\int_{a}^{b} f (x, {\bar{y}}_{s}) . d x) Δ y_{r} ≃ \sum_{s = 1}^{m} g ({\bar{y}}_{s}) Δ y

taking the limit $m \to \infty$ :

I = \int_{c}^{d} (\int_{a}^{b} f (x, y) . d x) . d y

eg: evaluate $I = \int \int_{R} x y . d x . d y$ , where $R$ is the rectangle with vertices $(0, 0)$ , $(0, 2)$ , $(1, 0)$ , $(1, 2)$
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