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# Investigating ratios of areas and volumes

Extracts from this document...

Introduction

Baca Gonzalez, Gabriel Alejandro        2106-005 May 2009

INVESTIGATING RATIOS OF AREAS AND VOLUMES

Introduction

The objective of this portfolio assignment is to investigate the ratio of the areas formed when  is graphed between arbitrary parameters  and  such that . This investigation may lead to a conjecture which ends up in a general formula.

Given the function , we can consider a region formed by this function from  and . The area between the function and the x-axis will be labeled B. The area from  to  and the y-axis will be labeled A.

[1]

The formula used to find the area under a curve to the x-axis is . To find the area to the y-axis the formula used is .

The area between the function and the x-axis is given by:

Therefore area of B is equal to .

So as the unit area is 1, the area of A is given by:

Another method which may be used is by getting the inverse of the function such that if , the inverse would be .

Therefore, the ratio of the areas A and B in the function  is   simplified into

Middle

2:1

1

2

3

15

3:1

0

2

2

8

2:1

0

2

3

16

3:1

0

3

2

27

2:1

n

A

n:1

Procedure

Area B

Area A

Graphing

For this section, the area of A was given by the inverse of the formula so that if  is equal to B,  is equal to A. Therefore, based in the results of the table, the conjecture remains as n:1.

The conjecture is true for the general case  from  and  such that  and for the following regions:

Area A: , , and the y-axis.

For example, when getting the area B from  to  with the function  is necessary to use . So, to find the area of the curve to the y-axis (area A) it is necessary to alter the limits so that they are  and . So as  and ,  and .

Until now the investigation has been working with ratios of areas; but what happens when dealing with volumes? In this final section, I will

Conclusion

x would be:

Let’s prove the conjecture with a random example:

So, by the use of the formula the result would be this:

And by the using of the conjecture, the result gives this:

Therefore, the conjecture for volumes of revolution generated by region B to the x-axis is correct.

Now, is there a general formula for ratios of volumes of revolution generated by region A to the y-axis?

Foe the development of this conjecture, I will use the same examples used before to find a conjecture for the x-axis.

So, as we can see the area now rotates around the y-axis, as the region A is the one that it is spinning now.

As it may be observed, as the power of the function is increased, the volume gets wider. This is opposed to the volume of revolution formed to the x-axis, which gets narrower when the exponent increases. Those two situations are connected, as when the volume of y gets wider, limits the space of the other volume and vice versa.

Now, the limits of the examples will be changed.

,

[1] All graphs were developed on Graph software, version 4.3.

[2] Plotting for volumes of revolution was graphed on Winplot.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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