- Level: International Baccalaureate
- Subject: Maths
- Word count: 919
IB Math HL Portfolio Type 1 (Ratio)
Extracts from this document...
Introduction
Area A formed in the function y=x2 from y=0 to y=1 and the y-axis can be found by integrating the function with respect to y.
Area A:
y=x2
x=
x=
dy
=[]
=[]
=(
= units2
Area B formed in the function y=x2 from x=0 to x=1 and the x-axis can be found by integrating the function with respect to x.
Area B:
y=x2
dx
=
=
=
= units2
Therefore, the ratio of the areas A and B for the function y=x2 is:
A : B = :
= 2 : 1
Now try the same with the function y=xn when nZ+ and between x=0 and x=1.
Area A formed in the function from y=0 and y=1n and the y-axis can be found by integrating the function in respect for y.
Area A:
y=xn
x=
dy
=[]
=[]
=(
= units2
Area B formed in the function from x=0 and x=1 and the x-axis can be found by integrating the function in respect of x.
Area B:
y=xn
dx
=
=
=
= units2
Middle
= 47 : 1
In y=x47, the value of n is 47, and therefore the conjecture n : 1 works when n=47.
When n=113,
Area A:
y=x113
x=
dy
=[]
=[]
=(
= units2
Area B:
y=x113
dx
=
=
=
= units2
Therefore, the ratio of the areas A and B in the function y=x113 is:
A : B = :
= 113 : 1
In y=x113, the value of n is 113, and therefore the conjecture n : 1 works when n=113.
It can be assumed that the conjecture n : 1 works in any positive real numbers.
- To determine whether the conjecture only holds for areas between x=0 and x=1, now try the same thing but changing the values of x.
When x is from 0 to 2,
Area A:
y=xn
x=
dy
=[]
=[]
=(
= units2
Area B:
y=xn
dx
=
=
=
= units2
Therefore, the ratio of the areas A and B when x is between 0 and 2 is:
A : B = :
= n : 1 ………. Conjecture works
When x is from 1 to 2,
Area A:
y=xn
x=
dy
=[]
=[]
=(
=
= units2
Area B:
y=xn
dx
=
=
=
=
= units2
Conclusion
A : B = :
= n : 1 ………. Conjecture works
s.
- When y=xn, x is between a and b and y is between an and bn,
Area A:
y=xn
x=
dy
=
=
=(
=
=n( units2
Area B:
y=xn
dx
=
=( units2
Therefore, the ratio of the areas A and B is,
A : B = n( : (
= n : 1 ………. Conjecture works
The conjecture n : 1 is true for the general case y=xn from x=a to x=b where a<b and the area A is defined as y=xn, y=an, y=bn and the y-axis and the area B is defined as y=xn, x=a, x=b and the x-axis.
- (a) We need to find the volume B in order to find out the volume A so we start off with volume B.
Volume B:
y=xn
dx
=
=
= units3
And then volume A.
Volume A:
=
=
=
=
=
= units3
Therefore, the ratio of the volumes A and B is:
A : B = :
=2n : 1
(b)Volume A:
y=xn
x=
dy
=
=
=
= units3
Volume B:
=
=
=
=
=
= units3
Therefore, the ratio of the volumes A and B is:
A : B = :
= n : 2
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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