HL- Type 1

INVESTIGATINGRATIOS OF AREAS AND VOLUMES

The purpose of this portfolio is to investigate the ratios of areas and volumes when a function y= xn is graphed between two arbitrary parameters x=a and x=b such that a‹b.

Task 1

The general formula to find area A is [pic]

The general formula to find area B is [pic]

Therefore, the ratio of Area A to Area B is-

= [pic] ÷ [pic]

= [pic] × [pic]

= n : 1

n:1 is the general conjecture formed.

The given function is in the form of y=xn. The function is y=x2. As mentioned above the parameters are between x=a and x=b. Here a=0 and b=1.

The graph of the function-

[pic]

y=xn where n=2.

The Area of the shaded region or area B was found to be 0.3333333333 units2. Area A = 1- Area B

= 1- 0.33333333

= 0.6666666666 units2

Mathematically the area can also be found by the following-

Area A can be found by the formula- [pic]

Area B can be found by the formula- [pic]

Total Area = 1 unit2

Area B = [pic]

= [pic]

Area A = [pic]

= [pic]

Ratio of area A : area B

[pic] : [pic]

= 2 : 1(satisfies the conjecture n:1)

Now I will investigate the ratios of area A and B with other natural numbers.

This means that the function will stay y=xn but here only n will change. The parameters will stay the same a=0 and b=1.

The ratios when n =3

The graph of y=x3

[pic]

Y=xn where n=3.

The Area of the shaded region or area B was found to be 0.25 units2. Area A = 1- Area B

= 1- 0.25

= 0.75 units2

Mathematically the area can also be found by the following-

Total area = 1unit2

Area B = [pic]

= [pic]

Area A = [pic]

= [pic]

Ratio of area A : area B

[pic] : [pic]

= 3 : 1(satisfies the conjecture n:1)

Now I will investigate the ratio of the areas for n being a rational number. The ratio when n= [pic]

Y = x0.5

[pic]

The Area of the shaded region or area B was found to be 0.66666666666 units2. Area A = 1- Area B

= 1- 0.66666666666666

= 0.33333333333333 units2

Mathematically the area can also be found by the following-

Total area = 1unit2

Area B = [pic]

= [pic]

Area A = [pic]

= [pic]

Ratio of area A : area B

[pic] : [pic]

= 0.5 : 1 (satisfies the conjecture n:1)

Observations from task 1-

• For the increasing values of n area A increases and area B decreases. • Area A can be found by the formula [pic]

• Are B can be found by the formula [pic]

• The ratio of areas A to B seem to be forming a general conjecture n: 1.

• The conjecture was even proving true for fractional values of n.

Task 2

Here I will also investigate the same function y=xn with parameters x=a and x=b. However the values of a and b will be different here. The new parameters are a=0 and b=2

When n=2

The graph of the function y=x2

[pic]

The Area of the shaded region or area B was found to be 2.66666666 units2. Area A = 8- Area B

= 8- 2.666666666666

= 5.33333333333 units2

Mathematically the area can also be found by the following-

Total Area = 8 units2

Area B = [pic]

= [pic]

Area A = [pic]

= [pic]

Ratio of area A : area B

[pic] : [pic]

= 2 : 1 (Satisfies the conjecture n:1)

When n=3

The graph of the function y=x3

[pic]

The Area of the shaded region or area B was found to be 4 units2. Area A = 16- Area B

= 16-4

= 12 units2

Mathematically the area can also be found by the following-

Total Area = 16 units2

Area B = [pic]

= [pic]

Area A = [pic]

= [pic]

Ratio of area A : area B

= 12 : 4

= 3 : 1 (Satisfies the conjecture n:1)

When n=1.5

The graph of function y= x1.5

[pic]

The Area of the shaded region or area B was found to be 2.262742...