5.5 Area of regions in the plane  (Page 2/3)

If $S$ is a closed geometric set, we will indicate the corresponding open geometric set by the symbol $S^0.$

The symbol $S^0$ we have introduced for the open geometric set corresponding to a closed one is the same symbol that we have used previously for the interior of a set.Study the exercise that follows to see that the two uses of this notation agree.

Now, given a geometric set $S$ (either open or closed), that is determined by an interval $[a,b]$ and two bounding functions $u$ and $l,$ let $P=\{x_0<x_1<...<x_n\}$ be a partition of $[a,b].$ For each $1\le i\le n,$ define numbers $c_i$ and $d_i$ as follows:

Because the functions $l$ and $u$ are continuous, they are necessarily bounded, so that the supremum and infimum above are real numbers.For each $1\le i\le n$ define $R_i$ to be the open rectangle $(x_{i-1},x_i)\times (c_i,d_i).$ Of course, $d_i$ may be $<c_i,$ in which case the rectangle $R_i$ is the empty set. In any event, we see that the partition $P$ determines a finite set of (possibly empty) rectangles $\left\{R_i\right\},$ and we denote the union of these rectangles by the symbol ${\mathcal{C}}_P.={\cup }_{i=1}^n(x_{i-1},x_i)\times (c_i,d_i).$

The area of the rectangle $R_i$ is $(x_i-x_{i-1})(d_i-c_i)$ if $c_i<d_i$ and 0 otherwise. We may write in general that $A\left(R_i\right)=(x_i-x_{i-1})max((d_i-c_i),0).$ Define the number $A_P$ by

Note that $A_P$ is not exactly the sum of the areas of the rectangles determined by $P$ because it may happen that $d_i<c_i$ for some $i$ 's, so that those terms in the sum would be negative. In any case, it is clear that $A_P$ is less than or equal to the sum of the areas of the rectangles, and this notation simplifies matters later.

For any partition $P,$ we have $S\supseteq {\mathcal{C}}_P,$ so that, if $A\left(S\right)$ is to denote the area of $S,$ we want to have

Let $S$ be a geometric set (either open or closed), bounded on the left by $x=a,$ on the right by $x=b,$ below by the graph of $l,$ and above by the graph of $u.$ Define the area $A\left(S\right)$ of $S$ by

$$A\left(S\right)=\underset{P}{sup}{A}_P=\underset{P=\{x_0<x_1<...<x_n\}}{sup}\sum _{i=1}^n(x_i-x_{i-1})(d_i-c_i),$$

where the supremum is taken over all partitions $P$ of $[a,b],$ and where the numbers $c_i$ and $d_i$ are as defined above.