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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 image00.pngimage00.png is graphed between arbitrary parameters image69.pngimage69.png and image70.pngimage70.png such that image71.pngimage71.png. This investigation may lead to a conjecture which ends up in a general formula.

Given the function image76.pngimage76.png, we can consider a region formed by this function from image01.pngimage01.png and image22.pngimage22.png. The area between the function and the x-axis will be labeled B. The area from image43.pngimage43.png to image64.pngimage64.png and the y-axis will be labeled A.

image74.jpg[1]

The formula used to find the area under a curve to the x-axis is image77.pngimage77.png. To find the area to the y-axis the formula used is image87.pngimage87.png.

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

image97.png

image103.png

image107.png

Therefore area of B is equal to image110.pngimage110.png.

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

image121.png

image130.png

image139.png

Another method which may be used is by getting the inverse of the function such that if image76.pngimage76.png, the inverse would be image153.pngimage153.png.

Therefore, the ratio of the areas A and B in the function image76.pngimage76.png is  image154.pngimage154.png simplified into image155.pngimage155.png

...read more.

Middle

2:1

1

2

3

15

image04.png

image05.png

3:1

0

2

2

8

image06.png

image07.png

2:1

0

2

3

16

image08.png

image09.png

3:1

0

3

2

27

image10.png

image11.png

2:1

n

A

image12.png

image13.png

n:1

Procedure

Area B

image14.png

image15.png

image16.png

image17.png

image18.png

image19.png

image20.png

image21.png

image23.png

image24.png

image25.png

image26.png

image27.png

image28.png

image29.png

image30.png

image31.png

image32.png

image33.png

image34.png

Area A

image35.png

image36.png

image37.png

image38.png

image39.png

image40.png

image41.png

image42.png

image44.png

image45.png

image46.png

image47.png

image48.png

image49.png

image50.png

image51.png

image52.png

image53.png

image54.png

image55.png

Graphing

image56.png

image57.jpg

image58.png

image59.jpg

image60.png

image61.jpg

image62.png

image63.jpg

image65.png

image66.jpg

For this section, the area of A was given by the inverse of the formula so that if image67.pngimage67.png is equal to B, image68.pngimage68.png 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 image00.pngimage00.png from image69.pngimage69.png and image70.pngimage70.png such that image71.pngimage71.png and for the following regions:

Area A: image00.pngimage00.png, image72.pngimage72.png,image73.pngimage73.png and the y-axis.

For example, when getting the area B from image22.pngimage22.png to image75.pngimage75.png with the function image76.pngimage76.png is necessary to use image67.pngimage67.png. 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 image64.pngimage64.png and image78.pngimage78.png. So as image79.pngimage79.png and image80.pngimage80.png, image81.pngimage81.png and image82.pngimage82.png.

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

...read more.

Conclusion

x would be:

image128.png

Let’s prove the conjecture with a random example:

image129.png

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

image131.png

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

image132.png

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.

image84.png

image133.png

image134.pngimage135.png

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.

image76.png

image136.png

image137.pngimage138.png

image92.png

image140.png

image141.pngimage142.png

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.

image143.png

image144.pngimage145.png

Now, the limits of the examples will be changed.

image79.pngimage79.png, image80.pngimage80.png

image108.png

image146.png

image147.pngimage148.png

image76.png

image149.png

image150.pngimage151.png

image92.png
image152.png


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

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

...read more.

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