Investigating ratios of areas and volumes

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Baca Gonzalez, Gabriel Alejandro        2106-005 May 2009



The objective of this portfolio assignment is to investigate the ratio of the areas formed when  is graphed between arbitrary parameters  and  such that . This investigation may lead to a conjecture which ends up in a general formula.

Given the function , we can consider a region formed by this function from  and . The area between the function and the x-axis will be labeled B. The area from  to  and the y-axis will be labeled A.

The formula used to find the area under a curve to the x-axis is . To find the area to the y-axis the formula used is .

The area between the function and the x-axis is given by:

Therefore area of B is equal to .

So as the unit area is 1, the area of A is given by:

Another method which may be used is by getting the inverse of the function such that if , the inverse would be .

Therefore, the ratio of the areas A and B in the function  is   simplified into

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But will this ratio remain as we increase the power of the function? This will be proven by elevating the exponent of the formula by 1 each time so this can lead to a conjecture.

Testing variables for n


Therefore, the ratio is given by the formula.


From the graphs we can observe that as the value of n keeps increasing the area between the curve and the x-axis gets narrower. By the other side, as n increases the area A increases too.

So as we can ...

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