Areas in the Plane

When discussing area, it is not possible to avoid the limit concept, and this brings a topic usually associated with Geometry into the field of Analysis. One could even make a case for including much of Geometry as a subtopic of Analysis since Geometry involves properties of distance, a distinguishing feature of Analysis.

What properties of area can be taken as given? One would hope that whatever axioms are chosen, they would let you prove results about area that you know to be true from Euclidean Geometry. The following axioms accomplish this.

Axioms for Area

1. The area of a set in the plane is a nonnegative real number.

2. A square with side length 1 has area equal to 1.

3. (Similarity) If sets A and B are similar in the geometric sense with lengths in B equal to t times the corresponding lengths in A, then the area of B is t² times the area of A.

4. (Area Zero) Let A be a set. Suppose that for each > 0 there is a sequence of squares S₁; S2; S3;    with areas s1; s2; s3;    , respectively, such that the set A is contained in the union of the squares ¹[

kD1Sk, and for every natural

Axioms 1, 2, and 3 should agree with what you know about area from Geometry, and they can be used to prove some simple results. For example, since a11 square has area 1, Axiom 3 can be used to show that an s  s square has area s².

The result from the previous section that a line segment has area 0 is particularly useful because of the way it can be used in conjunction with the Union area axiom.

In particular, suppose A and B are two squares or other polygons set side-by-side so that they only share an edge. Because the shared edge is a line segment, it has 0 for its area, and the Union area axiom shows that A [ B has an area equal to the sum of the area of A and the area of B. By using mathematical induction, this result can be extended to the union of many polygons that share borders. In particular, consider finding the area of a rectangle with width x and length y. If ^x_y is a rational number equal to^p_q, where p and q are positive integers, then the x  y rectangle is the union of p  q squares all with side length ^x_p. Indeed, the width of the rectangle which has length x is spanned by p such squares, and the length of the rectangle which has length y is spanned by q such squares showing that the entire rectangle can be tiled by a p  q array of squares, each with area

x p

₂

. The Union axiom then shows that the area of the x  y rectangle is p  q  the last of the area axioms to conclude that the area of any rectangle is equal to its length times its width even when the length of the rectangle is an irrational multiple of its width.

The last area axiom is essentially the Method of Exhaustion used some by Euclid and much more extensively by Archimedes to calculate areas and volumes.

It is an example of a use of Calculus about 1800 years before the foundation of Calculus was formally established by Newton and Leibniz. This axiom says that if a region in the plane can be closely approximated by sets whose areas you know, then you can figure out the area of the region. Take, for example, a rectangle B with

168 6 Riemann Integrals width x > 0 and length y > 0 where the ratio ^y_x D ˛ is irrational. It is certainly possible to find other rectangles close to the size of B whose length to width ratios are rational. To prove that B has area xy, the axiom requires that for each > 0 you find a subset A  B whose area is greater than xy  and a set C containing B whose area is less than xy C. Suppose you choose A to be a rectangle with width x and length just a bit short of y, say rx, where r is a rational number chosen to be less than but suitably close to ^y_x. How close is suitably close? Well, you would need the area of A, which is x  rx D rx², to be within of xy, that is, xy  rx² < . Solving for r shows that r> ^y_x_x^₂. Is there such an r which is rational and between^y_x_x^₂ and^y_x? Of course there is. The rational numbers are dense in the real line; there are rational numbers in every interval of positive length. Thus, you can select a rational number r between ^y_x  _x^₂ and ^y_x and let A be an x  rx rectangle. Then A can be placed inside of B, and the area of A is within of xy. Similarly, you can choose a rectangle C with width x and length sx, where s is a rational number chosen to be greater than but suitably close to ^y_x. You need the area of C to be within of xy, so choose s so that x  sx  xy< . This happens if ^y_x < s < ^y_x C _x^₂. Since you have found a rectangle A contained inside B and a rectangle C containing B with the areas of A and C within of xy, the Exhaustion area axiom shows that B has area xy.

The familiar formula for the area of a triangle given as one half the base times the height can be derived geometrically, but to prove this formula using the area axioms requires more work. To begin, consider a right triangle with legs with lengths x and y. Place this triangle in a rectangle with side lengths x and y. For any natural number n, the rectangle can be overlaid with an n  n grid of rectangles with side lengths

x

n and ^y_n. The hypotenuse of the triangle is the diagonal of the x  y rectangle and spans the diagonals of n of the smaller rectangles as shown in Fig.6.5exhibiting the case where n D8.

Because there are n grid rectangles along the hypotenuse of the triangle, it must be that there are ⁿ²₂^n grid rectangles inside the triangle with a total area of ⁿ²₂^n _n^xy₂ D 

1 ¹_n_xy

2. Similarly, the triangle is enclosed inside the union of

n²Cn

2 grid rectangles with a total area of

1 C¹_n_xy

2. Clearly, n can be chosen large

Fig. 6.5 An8  8 grid of rectangles overlaying a triangle

enough to make both the total area of grid rectangles inside the triangle and the total area of grid rectangles enclosing the triangle within a particular > 0 of ^xy₂. Thus, the Exhaustion axiom shows that the area of the triangle is^xy₂ as expected. Since any triangle can be partitioned into two right triangles, the well-known area formula for the area of a triangle follows. Since any polygon can be partitioned into triangles, the usual formulas from Geometry for the areas of polygons can be derived in the same way they would be in Geometry.

You may wonder whether these techniques can be used to find the area of any region in the plane, or at least any bounded region in the plane. This is a really good question with a very complicated answer. The Area Axioms listed in this section are designed to give the reader a feel for proofs about areas that will be useful in the upcoming discussion of proofs about Riemann integrals. The axiom list is not complete enough to allow the calculation of the area of many of the sets that one might encounter. The area of Analysis known as Measure Theory provides a somewhat richer environment for this study, but the complexities of measure theory go beyond the aim of this text. What can be said is that even with the use of measure theory, there are sets in the plane complex enough that one cannot assign an area measure to them.

6.4.1 Exercises

1. Show that a circle with radius r has arear².

2. Suppose the polygonal region A in the coordinate plane has area K. Show that the region f.x; y/ j .x;^y₃/ 2 Ag has area 3K.