# 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 and the x-axis between the limits x=a and x= b. B on the other hand is the area defined by And the y-axis between the limits and . 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, . 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 , and the y-axis from To which are y = 0 and y = 1. The graph below shows the graph of With the areas of A and B shaded respectively

Graph of

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

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:

Middle

Value of n | Area of A | Area of B | Ratio of A to B |

2 | 2 | ||

3 | 3 | ||

4 | 4 | ||

5 | 5 | ||

10 | 10 | ||

18 | 18 | ||

29 | 29 | ||

35 | 36 | ||

41 | 41 | ||

47 | 47 | ||

52 | 52 | ||

61 | 61 | ||

64 | 64 | ||

69 | 69 | ||

71 | 71 | ||

81 | 81 | ||

93 | 93 | ||

100 | 100 | ||

1000 | 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 .

Area of B

Area of A

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 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 With the Area of A shaded

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.

Conclusion

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 from x=a to x= b.

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

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

First example:

when a=4 and b= 8

Second example:

When a=2 and b=4

Third example:

When a=1 and b=2

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 , 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.

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

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