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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