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Investigating ratio of areas and volumes

Extracts from this document...

Introduction

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 = xnis 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 = xnis graphed between two arbitrary parameters x = a and x=b such that a<b will be investigated.

image00.png

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/nwe the following is obtained:


image01.png

...read more.

Middle

y = xn and the x-axis.

However, to find Volume A using modern technology is tricky.

Volume A can be expressed in another manner. Consider the horizontal line that meets the graph at the point x = b (y = bn):


image02.png

Volume A can be considered as the volume contained around the x-axis by the graph of this horizontal line (y = bn)between the points x = 0 and x = b subtracted by volume B and the volume contained around the x-axis by the graph of y = an between the points x = 0 ans x = a. This last volume will be considered as volume C.

This method will be used to find the ratio Volume A: Volume B.

Investigation:

y = x2between x = 0and x = 1:

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

Volume B will equal the volume contained around the x-axis between the graph and the x-axis between the points x = 0 and x = 1.

Volume A will equal the volume contained around the x-axis between the graph of  y = 12 (= 1) and the x-axis between the points x = 0 and x = 1 subtracted by volume B. (Note: since in this case a = 0 volume C = 0).

Using the program Autograph 3.20 the following results are obtained:

Volume B = 0.2π

Volume under y = 1 = 1π

Volume A = 1 - 0.2π = 0.8π

Ratio volume B: volume A = 0.8π: 0.2π

= 4: 1

y = xnbetween x = 0and x = 1:

Now consider other functions of the type y = xn, n image03.png Z+ between the points x = 0 and x = 1.

Volume B will be the volume contained around the x-axis between the graph and the x-axis between the points x = 0 and x = 1.

Volume A will be the volume contained between the line y = 1n (= 1) and the x-axis between the points x = 0 and x = 1 subtracted by volume B (volume C = 0).

...read more.

Conclusion

y = 2 and the x-axis between the points x = 0n (= 0) and x = 2n subtracted by volume A (a = 0 therefore volume C = 0).

n

Area under horizontal line (in π)

Volume A (in π)

Volume B (in π)

Volume A / Volume B

2

16

8.00

8.00

1.0

3

32

19.20

12.80

1.5

4

64

42.65

21.35

2.0

5

128

91.35

36.65

2.5

6

256

191.70

64.30

3.0

7

512

397.50

114.50

3.5

8

1024

817.40

206.60

4.0

9

2048

1671.00

377.00

4.4

10

4096

3404.00

692.00

4.9

π

35.3

21.56

13.74

1.57

e

26.32

15.16

11.16

1.36

From these results it is evident that the conjecture holds true between the points x = 0 and x = 2.

y = xnbetween the points x = 1 and x = 2:

Volume A will equal the volume contained around the x-axis between the graph of y = x1/n and the x-axis between the points x = 1n (= 1)and x = 2n.

Volume B will equal the volume contained around the x-axis between the graph of y = 2 and the x-axis between the points x = 0 and x = 2n subtracted by volume A and volume C.

Volume C will equal the volume contained around the x-axis between the graph of y = 1 and the x-axis between the points x = 0 and x = 1n (=1).


n

Area under horizontal line (in π)

Volume A (in π)

Volume B (in π)

Volume C (in π)

Volume A / Volume B

2

16

7.5

7.5

1

1.0

3

32

18.6

12.4

1

1.5

4

64

42.66

20.34

1

2.1

5

128

91.41

35.59

1

2.6

6

256

191.9

63.1

1

3.0

7

512

398

113

1

3.5

8

1024

818.6

204.4

1

4.0

9

2048

1674

373

1

4.5

10

4096

3410

685

1

5.0

π

35.3

21.56

12.74

1

1.69

e

26.32

15.17

10.15

1

1.49

From these results it is evident that the conjecture holds true between the points x = 1 and x = 2.

y = xnbetween the points x = e and x = π:

Volume A will equal the volume contained around the x-axis between the graph of y = x1/n and the x-axis between the points x = enand x = πn.

Volume B will equal the volume contained around the x-axis between the graph of y = π and the x-axis between the points x = 0 and x = πn subtracted by volume A and volume C.

Volume C will equal the volume contained around the x-axis between the graph of y = e and the x-axis between the points x = 0 and x = en.


n

Area under horizontal line (in π)

Volume A (in π)

Volume B (in π)

Volume C (in π)

Volume A / Volume B

2

97.41

21.41

21.40

54.6

1.0

3

306

94.56

63.04

148.4

1.5

4

961.3

372.00

185.93

403.4

2.0

5

3020

1374.00

549.56

1097

2.5

6

9488

4881.00

1626.23

2981

3.0

7

29807

16880.00

4824.84

8102

3.5

8

93642

57300.00

14317.58

22024

4.0

9

294185

191700.00

42616.50

59869

4.5

10

924210

634600.00

126870.13

162740

5.0

π

359.9

115.40

73.50

171

1.57

e

221.7

63.19

46.51

112

1.36

From these results it is evident that the conjecture holds true between the points x = e and x = π.

Proving the conjecture:

Using calculus the conjecture can be proven.

Volume A:

image07.png


Volume B:

image08.png


Ratio volume A: volume B:

image09.png

...read more.

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