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Investigation of Area Under a Curve and Over a Curve

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Introduction

Quintero  

        In this investigation I will be examining the relationships of the area under a curve and over a curve  between two specified limits. Here I will call A the area surrounded by image00.png and the x-axis between the limits x=a and x= b. B on the other hand is the area defined by image00.png And the y-axis between the limits image66.pngand image75.png.  The vales of a and b will be whole real numbers. The investigation will test the ratio between A and B.

        Initially, I will begin by finding the ratio of A to B using a simple function, image27.png. The area of B is surrounded by this curve  and the x-axis from x = 0 to x = 1 (a=0 and b=1). The area of A will be surrounded by the curve image27.png, and the y-axis from image106.pngTo image117.pngwhich are  y = 0 and y = 1. The graph below shows the graph of image27.pngWith the areas of A and B shaded respectively

Graph of image27.png

image01.png

First I will find the area of B by using the formula of image11.png. This formula however translates into image19.png. In here a=0 and b=1 and image27.png.

image37.png

Now that I have calculated the area of B I will calculate the area of A, but to do so there are 3 possible methods that can be used.

Method 1:

...read more.

Middle

.

Value of n

Area of A

Area of B

Ratio of A to B

2

image108.png

image109.png

2

3

image110.png

image111.png

3

4

image112.png

image113.png

4

5

image114.png

image115.png

5

10

image116.png

image118.png

10

18

image119.png

image120.png

18

29

image121.png

image122.png

29

35

image123.png

image124.png

36

41

image125.png

image126.png

41

47

image127.png

image128.png

47

52

image129.png

image130.png

52

61

image131.png

image132.png

61

64

image133.png

image134.png

64

69

image135.png

image136.png

69

71

image137.png

image138.png

71

81

image139.png

image140.png

81

93

image141.png

image142.png

93

100

image143.png

image144.png

100

1000

image145.png

image146.png

1000

        After analyzing the table, I’ve noticed a pattern correlating between n and the ratio between A and B. From this I will infer that the value of n is equal to the ratio. However, to make my conjecture valid I will have to test it with values of n other than positive integers. My next step will be to investigate the relation between A and B when n is a negative integer.

The first equation I will test with a negative  integer for  n is image147.png.

Area of B

image02.png

Area of A

image03.png

The area for both A and B calculate out to be  undefined. In order to get a better understanding of the situation, I will graph the area of B and the area of A separately.

Graph image04.png With area of B shaded

This graph shows how the area of B is infinite since the function will go on into infinity and never touches the y axis.

Graph of image04.png With the Area of A shaded

image05.png

This graph shows that the area of A is also infinite like that of B. This supports the calculations done earlier because it shows that because the areas of both are undefined, they are infinite.

...read more.

Conclusion

image09.png When a=4 and b=8

image50.png

My original conjecture applies to these types of area because the a and b will make a proportion since the ratio is still equal to n.

        Finally I will prove my conjecture mathematically. The area of A will remain that space surrounded by the y-axis, the variation of the by image51.png from x=a to x= b.

image51.png

image53.png

image54.png

image55.png

Now, I will prove that the ratio between A and B is equal to n.

image56.png

        I will use actual numbers to plug into the equation and confirm that image57.png works.

First example:

image58.png when a=4 and b= 8

image59.png

Second example:

image60.pngWhen a=2 and b=4

image61.png

Third example:

image58.png When a=1 and b=2

image63.png

All of the equations above proved my conjecture right.

In total, from the entire investigation I can make the following reasoning:

When a curve follows the model of image64.png, the relationship between A (the area surrounded by the curve, the y-axis and the lines y=f(a) and y=f(b)) and B (the area surrounded by the curve, the x-axis and the lines x=a and x=b) will be that the ratio of A to B will be equal to the value of n. This applies to positive integers, positive rational, and real numbers(including irrational).  Limitations to this however are negative values of n for which I conclude the areas for both A and B will be undefined or infinite.

...read more.

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