Investigating Ratios of Areas and Volumes

Michael Zuber

Introduction:

This paper has been separated into three parts:

Part i investigates the ratio of the areas formed when y = xn is graphed between two arbitrary parameters x = a and x=b such that a<b.

Part ii investigates the ratio of the volumes formed around the x-axis when y = xn is graphed between two arbitrary parameters x = a and x=b such that a<b.

Part iii investigates the ratio of the volumes formed around the y-axis when y = xn is graphed between two arbitrary parameters x = a and x=b such that a<b.

To investigate these ratios the program Autograph 3.20 was used in order to obtain sets of results. All areas were found using Simpson’s rule at 50 divisions.

Part i:

Introduction:

In this investigation the ratio of the areas formed when y = xn is graphed between two arbitrary parameters x = a and x=b such that a<b will be investigated.

This investigation will investigate the ratio area A: area B.

Area A is the area contained in between the graph and the y-axis between the arbitrary parameters x = a and x = b.

Area B is the area contained in between the graph and the x-axis between the arbitrary parameters x = a and x = b.

However, in investigating this there is a small problem. Using modern technology the area between the graph and the x-axis can easily be found. However, not all programs allow for the area in between the graph and the y-axis to be found.

This problem, however, can be resolved. Area A can be expressed in another form. Considering the inverse function of y = xn, y = x1/n is found. Considering the graph of y = x1/n we the following is obtained:

In doing this, area A and area B are switched. Area A is now the area contained between the graph and the x-axis between the arbitrary points x = an and x = bn and area B is the area contained between the graph and the y-axis between the arbitrary points x = an and y = bn.

By this the ratio area A: area B can more readily be investigated.

Area A is the area contained in between the graph of y = x1/n and the x-axis between the two arbitrary points x = an and x = bn such that a<b.

Area B is the area subtended in between the graph of y = xn and the x-axis between the two arbitrary points x = a and x = b such that a<b.

Investigation:

y = x2 between x = 0 and x = 1:

Consider the graph y = x2 between the two points x = 0 and x = 1.

Area B will be the area contained between the graph of y = x2 and the x-axis between points x = 0 and x = 1.

Area A will be the area contained between the graph of y = x1/2 (= √x) and the x-axis between points x = 02 (= 0) and x = 12 (= 1).

Using the program Autograph 3.20 the following results are obtained:

Area A = 0.666

Area B = 0.333

Ratio area A: area B = 2:1

y = xn between x = 0 and x = 1:

Now consider other functions of the type y = xn, n Z+ between the points x = 0 and x = 1.

Area B will be the area contained between the graph of y = xn and the x-axis between the points x = 0 and x = 1.

Area A will be the area contained between the graph of y = x1/n and the x-axis between the points x = 0n (= 0) and x = 1n (= 1).

Below is a table for the results of area A, area B and the ratio area A / area B for the values of n between 2 and 10.

(Note: Results for area A / area B were rounded to 1 d.p. and values for Area A and Area B were rounded to 3 d.p.)

From the above data the following conjecture can be made:

For the graph of y = xn in between the points x = 0 and x = 1 the ratio area A: area B is:

n: 1.

Testing the conjecture:

The conjecture will be tested against the three real numbers π, e, and 100.