# IB Math HL Portfolio Type 1 (Ratio)

1.

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

Therefore, the ratio of the areas A and B for the function y=xn is:

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 =  :

= ...