Investigating ratios of areas and volumes
The ratio of the areas formed when y=x^n is graphed between two arbitrary parameters x=a and x=b such that a<b.
1. Given the function y=x^2, consider the region formed by this function from x=0 to x=1 and the x-axis. Label the region from y=0 to y=1 and the y-axis area A.
Find the ratio of area A: area B.
Calculate the ratio of the areas for other functions of the type y=x^n, nZ^+ between x=0 and x=1. Make a conjecture and test your conjecture for other subsets of the real numbers.
2. Does your conjecture hold only for areas between x=0 and x=1? Examine for x=0 and x=2, x=1 and x=2, etc.
3. Is your conjecture true for the general case y=x^n from x=a to x=b such that a<b and for the regions defined below? If so prove it; if not, explain why not.
Area A: y=x^n, y=a^n, y=b^n and the y-axis
Area B: y=x^n, x=a, x=b and the x-axis