Mathematics and Logic - Skill and Concept Development

Chapter 17. Area Approximation

Volume 3, Why Slopes and More Math.

Covering A Region by Squares

The areas of squares and rectangles may be calculated by calculating the product of the lengths of their sides. In the plane, the area of a bounded region $S$ rectangular or not, may be approximated by covering the region $S$ concerned with small squares, all of the same size, overlapping, if at all, only at their edges.

The example of an elliptic shaped region $S$ is shown. Each covering by small squares gives three methods for approximating what the area $A$ of the region should be.

1. An inner and lower/under approximation to the area $A$ of the region $S$ can be obtained by summing up the areas of all the squares contained completely in the region $S$. This inner approximation is expected to yield an estimate \em lower \rm or $\le A$.

2. An outer and upper/over approximation of the area can be obtained by summing up the areas of the squares which have an interior point in common with the region. This outer approximation is expected to yield an estimate higher or $\ge A$. A point in a square but not on an edge is said to be an interior point of the square.

3. A middle approximation might be obtained by adding to the inner approximation, the areas of those square which are completely in or more than half-in the region $S$. Other inbetween approximations are possible. Intermediate approximations yield an middle area estimate between the upper and lower estimates.

   From a computational perspective, more than half-in but not completely in is not easy to define. This could be a matter of visual judgment -- a step outside of the domain of rule-based mathematics. To give a mathematical algorithm, the toss of a coin might be sufficient, or a judgment could be made on how many of the four triangles formed by the diagonals are included completely in the region $S$.

One or more of the above approximations may be familiar to you from your elementary school days.

Each of the above approximations is expected to improve as the squares are quartered (their sides halved) repeatedly and indefinitely. The latter would cause the lower estimate to increase, the upper estimate to decrease while the middle estimate together with the area $A$ presumably approximate, remaining inbetween. Such halving results three sequences of numbers or quantities.

The lower estimates yield the increasing sequence, the upper estimate yield the decreasing sequence, and the middle estimates yield a sequence between the previous two.

The area $A$ should be the common, finite, limiting value $L$ of the approximations as the sides of the covering squares become smaller (approach zero). This says how to compute the area $A$ with an unlimited accuracy if a common, finite limiting value $L$ exists for the approximations.

The area of a region is defined by the methods for approximating it. That is, the region has an area $A=L$ if and only if the three numerical approximations described above all approach a single finite limiting value $L$. This limiting $L$ is then called the area of the region. Otherwise, with some disappointment perhaps, we may say that the area is not defined. (Alternatively, we might define inner and outer areas using the limiting values of the inner and outer approximations and identify circumstances in which they are equal.

Optional (Advanced Material: For some odd (pathological) region, inner and outer area approximations may approach an inner limit $L_{inner}$ and an outer limit $L_{outer}$ which are not equal. Regardless, these limiting values may define what is meant by inner or outer concepts of area. There has been a concern with identifying conditions in which inner and outer approximations etc approach the same limits. For more information, a very specialized (that is, not for everyone) book or course on area computation, more precisely, integration theory, is indicated.

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