FINDING AREAS UNDER CURVES WITH HAND-HELD CALCULATORSAuthor(s): ARTHUR A. HIATTSource: The Mathematics Teacher, Vol. 71, No. 5, Computers and Calculators (MAY 1978), pp.420-423Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27961289 .

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FINDING AREAS UNDER CURVES WITH HAND-HELD CALCULATORS

No, this is not the usual limit of the sum from calculus. All you need are some points on

the curve?and, of course, your calculator.

By ARTHUR A. HIATT California State University-Fresno

Fresno, CA 93740

While the debates about the hand-held calculator go on, perhaps its real signifi cance is being overlooked. Clearly, we now have a tool that can assist mathematics educators to focus on one of the most ne

glected areas of mathematics instruction, that is, the method of inquiry used in

mathematics. We tend to overemphasize content, whereas an equally important as

pect of mathematics is its method of in

quiry. As a minimum, mathematics instruction

should help the student develop the abil

ity?

1. to make observations

2. to organize observations (data) a) to recognize patterns b) to make conjectures

3. to specialize and generalize a) to use inductive reasoning b) to reason by analogy

4. to invent symbolism to express mathe matical ideas

a) to accept conventional symbolism 5. to prove conjectures

a) to invent or accept an axiomatic structure.

It is the intent of this paper to apply the method of inquiry above to a problem in

secondary school geometry?the area of a circle. Often the approach is quite intuitive, showing that the area of a regular inscribed

polygon approaches the area of a circle as the number of sides of the polygon in creases (Jacobs & Myer 1972, p. 595). Since

the calculation of the area of a regular poly gon becomes increasingly tedious as the number of sides increases and requires trig onometry, most teachers rely on the stu dents' intuitive idea of limits. In general, the student has to accept the fact that the circumference of a circle is given by 2irr.

A more primitive, but perhaps more in

structive, method is to trace circles of vari ous radii on graph paper and use a simple balance to weigh them. I have used this method with secondary general mathemat ics students to determine the constant in

A = kr1 (formula for the area of a circle). The largest average value of k was 3.34. This value was obtained with a balance made from a straw and three pins. One pin is used as a fulcrum, one holds the cut-out circle, and one holds the comparable amount of

squared paper (figure 1). A value of k using a simple commercial balance was found to be close to 3.1. This same method was used

by Galileo in about 1600 to predict the formula for the area of a cycloid (Eves 1969, p. 295). Galileo conjectured the area

of a cycloid to be about three times the area of the generating circle. The first published proof that the area is exactly three times the

generating circle was in 1644 by Evangelista Torricelli, a student of Galileo. The method of proof involved the use of infinitesimals

(the early beginnings of integral calculus).

Fig. l

420 Mathematics Teacher

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It is informative to recapitulate the

method of Galileo, for it emphasizes the

method of inquiry used in mathematics.

First he made several observations about

the area of cycloids (weighings of various

cycloids generated by circles of various

radii). After carefully organizing his data, he conjectured that the area of a cycloid is

approximately three times the area of the

generating circle (Ac ? 3 *). Finally, new

axioms and definitions had to be developed to construct a formal proof. His student,

Torricelli, gave us this proof. Thus we see

how content (Ac =

Inr2) is added to the

body of mathematical knowledge. Clearly, the method of discovery is in many cases as

exciting as the actual fact (content), if not

more so.

Let us return to the area of a circle. The

appendix indicates how to calculate the

area of polygons, given the coordinates of

the vertices (Hiatt 1972, p. 598). The area of

a polygon is given by

n_i

(1) A = y [ + -ytXt + i)

+ xnyi -

ynXl

where starting with any vertex (xu yx) the

other vertices are numbered (x2, y2), etc., in

a. counterclockwise manner. As an ex

ample, consider figure 2. What is the area

of A BCD?

By equation (1)

A =

JK7.5)

Fig. 2

Clearly the area is 4 X 4 = 16 square units.

7-5 + 7-5 + 3? 1 + 3? 1

- 1-7

- 5-3

- 5-3

- 1-7

= -(76

- 44)

= 16.

We now have a powerful method to de

termine the area under any curve. Suppose we want the area under the curve as in

figure 3.

Fig. 3. PR is one-fourth the circle of radius 10.

Using a hand-held calculator, we can gen erate many points on x2 + y2

= 102, and by

applying equation 1 for the area of a poly gon, we can approximate the area under the curve. The product of this area and four is

approximately 100 square units. With the

points (0, 0), (10, 0), (9, 4.36), (8, 6), (7, 7.14), (6, 8), (5, 8.66), (4, 9.16), (3, 9.54), (2, 9.80), (1, 9.95), and (0, 10), we get 310

square units for an approximation to the area of the circle. This is equivalent to find

ing the area of a forty-sided polygon in

scribed in the circle. But with a hand-held calculator it is easy to get many ordered

pairs, including the following, (9.5, 3.12),

(8.5, 5.27), (7.5, 6.61), (6.5, 7.60), (5.5, 8.35), (4.5, 8.93), (3.5, 9.37), (2.5, 9.68), (1.5, 9.89), and (0.5, 9.99), If a polygon is

May 1978 421

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drawn that connects all twenty-two of the

preceding coordinates, its area is 78.2

square units. Therefore, the approximate area of the circle is 313 square units.

In a class with more than one calculator

available, students could complete table 1.

TABLE 1

Area of approximating Radius (r) polygon Area -r r*

10 (40 sides) 310 3.10 10 (80 sides) 313 3.13

More enterprising classes may want to

investigate the average length of segments from the origin to the perpendicular bisec tor of the sides of the polygon. Obviously, the hand-held calculator is a very useful

tool; without it, the data for the above ex

periment would be very difficult to gener ate, because of the tedious calculations nec

essary. Such a demonstration usually convinces students that the area of a circle is Trr2. This helps them accept the intuitive limit argument commonly used in geome try.

The reader may want to experiment with this method to see how really powerful it is. For example, the area under y

= 2 between = 0 and = 5 is 41.67 square units. How

many points on y = 2 must be taken to get

a polygon whose area differs from 41.67 by less than 1 percent?

APPENDIX

Areas from the Coordinates of the Vertices

We develop a formula for the area of a

triangle when the coordinates of its ver

tices are given in a coordinate plane. Since

any convex polygon can be divided into a

finite number of triangles, the method can

be generalized to include all convex poly gons.

Let triangle ABC be given as below and

compute the areas as shown:

area AABC = area ABFG

- area AG AC - area ACBF

= \{AG + BF)(GC + CF) -

?-AG-GC -

h-CF BF

= \(AG-CF + BFGC) = *[(/

- b)(c - e) + (/

- d)(e - a)} =

h(cf + eb + ad ? de ? aj ?

be).

Notice the pattern: three positives and three negatives. If we list the ordered pairs of the vertices in the following manner

and form the products as indicated, we get a formula for the area of AABC :

422 Mathematics Teacher

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c d?

area A ABC = -

= Kc/ + + ad ? ed ? a/

? c6).

The rule is simple. Start at any ordered

pair. Go in a counterclockwise direction

and affix the first ordered pair to the end

of the array. In our example, (c, d) ap

pears at the top and at the bottom. If we

had an n-gon, we would have (n + 1) ordered pairs in our array; the procedure for multiplying remains the same. The reader may enjoy proving the formula for

the n-gon.

e f a b

c d

REFERENCES Eves, Howard. History of Mathematics. New York:

Holt, Rinehart & Winston, 1969.

Hiatt, Arthur. "Problem Solving in Geometry." Mathematics Teacher 65 (November 1972):595-600.

Jacobs, Russell, and Richard Meyer. Discovering Ge

ometry. New York: Harcourt Brace Jovanovich,

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May 1978 423

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