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Investigating Ratios of Areas and Volumes

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Introduction

Devan

Bob Devan

Calculus AB/BC

7th & 8th

December 6, 2008

Investigating Ratios of Areas and Volumes

1.Given the function y=, consider the region formed by this function from =0 to =1 and the -axis. Label this area B. Label the region from y=0 to y=1 and the y-axis area A.

A. Find the ratio of area A: area B.

The area of A for the given function of is.

Area A

The area of B for the given function of is

Area B:

Thus the ratio of area A : area B can be given as 2:1.

B. Calculate the ratio of the areas for other functions of the type between 0 and . Make a conjecture and test your conjecture for other subsets of the real numbers.

Area A: 1-

Area B:

Conjecture: Given the function

Middle

Area A:

The area of B for the function is

Area B:

Ex 3)

Area A:

Area B:

3. Is your conjecture true for the general case from to such that and for the regions defined below? If so prove it; if not explain why not.

Area A:and the y-axis

Area B: and the x-axis

Area A: = = =

Explanation: The equation of can be stated in terms of y as or . The integral for the area of A is put into terms of y and then solved.

Area B: =

Explanation: Instead of putting the integral in

Conclusion

π = =

Explanation: Region B has no hole in the center thus it can be integrated together without separation.

Conjecture: The ratio of volumes of Region A to Region B is 2n:1. This can be proven by taking the volume of Region A and dividing it by the volume of Region B.

Region A around y-axis

Explanation: Since region A around the y-axis is a solid figure there is no need to have multiple integrals. So the volume of region A is simply found by using the disk method.

Region B around y-axis

= =

Explanation: Since region B contains a portion cut out, to find the area of B we use the shell method.

Conjecture: My conjecture for the volumes of regions when rotated about the y-axis is the ratio of n:2. This can be proven by taking region A and dividing it by region B.

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

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