# Investigating ratios of areas and volumes

Baca Gonzalez, Gabriel Alejandro        2106-005 May 2009

INVESTIGATING RATIOS OF AREAS AND VOLUMES

Introduction

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

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

Procedure

Therefore, the ratio is given by the formula.

Graphing

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 ...