The graphs of these functions are shown below:
After considering several examples of the function of the type y = xn, between
x = 0 and x = 1, a clear pattern emerges. The pattern for the ratio seems to be n:1 or just n.
Proof of Conjecture:
= = −
B =
A = 1 − = − =
A: B = : = n:1 = n
To test the conjecture, I will now consider other values of n by including other subsets of real numbers. The chart below shows functions of the type y = xn, between x = 0 and x = 1.
The results below show that the conjecture holds for other subsets of real numbers.
The graphs of these functions are shown below:
One important subset of real numbers has been excluded from further consideration is negative numbers. This is because no corresponding y-value exists at x = 0. There is a vertical asymptote at x = 0 and a horizontal asymptote at y = 0, as they approach infinity. Thus y = x-n cannot be integrated as infinity is not a valid area. The graph below demonstrates them phenomena for
y = x-2:
To further test the conjecture, I will now consider the areas between other x-values. I will examine if the conjecture holds for x = 0 and x = 2, x = 1 and x = 2, and x = 2 and x = 5.
First, I will consider functions of the type y = xn, between x = 0 and x = 2. The conjecture seems to hold for the bounds x = 0 and x = 2.
Area of A and B is now defined as the following:
Area A: y = xn, y = an, y = bn, and the y-axis
Area B: y =x, x = a, y = b, and the x-axis
The graph below is an example of the bounds for y = x2 from x=1 to x=2
Next, I will consider functions of the type y = xn, between x = 1 and x =2. The conjecture seems to hold for the bounds x = 1 and x = 2.
Finally, I will consider functions of the type y = xn, between x = 2 and x =5. The conjecture seems to hold for the bounds x = 2 and x = 5.
After testing the conjecture for y = xn for different types of real numbers and different bounds, the conjecture still holds. Now, I will consider the conjecture in more general terms. For the general case y = xn from x = a to x = b such that a < b and for the regions defined below:
Area A: y = xn, y = an, y = bn, and the y-axis
Area B: y =x, x = a, y = b, and the x-axis
Proof:
B = = =
The formula for area A can be defined as:
= = = − =
Thus the ratio of A: B is n:
== n
Ratios of Volume
After considering and proving that the ratio of the area A: area B is simply n, I will now consider a general formula for the ratios of the volumes of revolution generated by the regions A and B when they are rotated about the x-axis and y-axis.
Proof for x-axis conjecture:
I will use the disc method to solve for volumes of revolution: V = , y is the function
y = xn
VB = = = = =
VA = = =
Thus ==
Proof for the y-axis:
VA = ==== =
VB = ===
= =
Conclusion
Thus the ratio of area A: area B is for the x-axis and for the y-axis.
