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# 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.

1. 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.

1. 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.

1. (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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