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# Investigation of Area Under a Curve and Over a Curve

Extracts from this document...

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

When a=4 and b=8

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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