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CaptainK
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Need to investgate the ratio of the areas formed when y=xn is graphed between two arbirary parameters x=a and x=b such that a<b
1. Given the funtion y=x2, consider the region formed by this function from x=0 to x=1 and the x-axis. Label this area B. Label the region form y=0 to y=1 and the y-axis area A.
Find the ratio of area A : area B
now this is where i had trouble
Calculate the ratio of the ares for other functions of the type y=xn, n [tex]\in[/tex] Z+ between x=0 an x=1. (not to complex but) 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=xn from x=a to x=b such that a<b and for the regions defned below? If so prove it; if not explain why not.
Area A: y=xn, y=an, y=bn and the y-axis
Area B: y=xn, x=a, x=b and the x-axis
4. Are there general formulae for the ratios of the volumes of revolution generated by the regions A and B when they are each rotated about
a) the x-axis?
b) the y-axis?
State and prove your conjecture
I can do the very first part of this rather easily but I'm having trouble making the conjcture because although it doesn't have to be correct, I do have to prove it later on. Questions 3 and 4 are the ony real problems so ny help with this would be of a major assistance to me. Much Thanks
1. Given the funtion y=x2, consider the region formed by this function from x=0 to x=1 and the x-axis. Label this area B. Label the region form y=0 to y=1 and the y-axis area A.
Find the ratio of area A : area B
now this is where i had trouble
Calculate the ratio of the ares for other functions of the type y=xn, n [tex]\in[/tex] Z+ between x=0 an x=1. (not to complex but) 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=xn from x=a to x=b such that a<b and for the regions defned below? If so prove it; if not explain why not.
Area A: y=xn, y=an, y=bn and the y-axis
Area B: y=xn, x=a, x=b and the x-axis
4. Are there general formulae for the ratios of the volumes of revolution generated by the regions A and B when they are each rotated about
a) the x-axis?
b) the y-axis?
State and prove your conjecture
I can do the very first part of this rather easily but I'm having trouble making the conjcture because although it doesn't have to be correct, I do have to prove it later on. Questions 3 and 4 are the ony real problems so ny help with this would be of a major assistance to me. Much Thanks
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