A : B = :
= n : 1
Conjecture for the ratio of the areas A and B for the function y=xn is A : B = n : 1.
Now, try and test the conjecture for other subsets of the real numbers.
When n=3,
Area A:
y=x3
x=
dy
=[ ]
=[]
=(
= units2
Area B:
y=x3
dx
=
=
=
= units2
Therefore, the ratio of the areas A and B in the function y=x3 is:
A : B = :
= 3 : 1
In y=x3, the value of n is 3, and therefore the conjecture n : 1 works when n=3.
When n=10,
Area A:
y=x10
x=
dy
=[ ]
=[]
=(
= units2
Area B:
y=x10
dx
=
=
=
= units2
Therefore, the ratio of the areas A and B in the function y=x10 is:
A : B = :
= 10 : 1
In y=x10, the value of n is 10, and therefore the conjecture n : 1 works when n=10.
When n=47,
Area A:
y=x47
x=
dy
=[ ]
=[]
=(
= units2
Area B:
y=x47
dx
=
=
=
= units2
Therefore, the ratio of the areas A and B in the function y=x47 is:
A : B = :
= 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
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 -10 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 -15 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 10 to 21,
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 91 to 214,
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
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
