5. Area

In chapter 2 we calculated the area of the set

$$\{(x,y)\in\mbox{{\bf R}}^2\colon 0\leq x\leq a \mbox{ and } 0\leq y \leq x^2\}$$

where $a \geq 0$, and of the set

$$\{(x,y)\in\mbox{{\bf R}}^2\colon 1\leq x\leq b \mbox{ and } 0\leq y \leq x^{-2}\}$$

where $b > 1$.

The technique that was used for making these calculations can be used to find the areas of many other subsets of $\mbox{{\bf R}}^2$. The general procedure we will use for finding the area of a set $S$ will be to find two sequences of polygons $\{I_n\}$ and $\{O_n\}$ such that

$$I_n\subset S\subset O_n \mbox{ for all } n\in\mbox{${\mbox{{\bf Z}}}^{+}$}.$$

We will then have

$$\mbox{\rm area}(I_n)\leq\mbox{\rm area}(S)\leq\mbox{\rm area}(O_n) \mbox{ for all } n\in\mbox{${\mbox{{\bf Z}}}^{+}$}.$$ (5.1)

We will construct the polygons $I_n$ and $O_n$ so that $\mbox{\rm area}(O_n)-\mbox{\rm area} (I_n)$ is arbitrarily small when $n$ large enough, and we will see that then there is a unique number $A$ such that

$$\mbox{\rm area}(I_n)\leq A\leq\mbox{\rm area}(O_n) \mbox{ for all } n\in\mbox{{\bf Z}}^+.$$ (5.2)

We will take $A$ to be the area of $S$.