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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. 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: 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): 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 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).

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: Volume B: Ratio volume A: volume B: This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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