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Math Help - Investigating ratios of areas and volumes

  1. #1
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    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.

    Investigating ratios of areas and volumes-calculus.jpg

    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.
    [IMG]file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/moz-screenshot.jpg[/IMG]
    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.

    Investigating ratios of areas and volumes-calculus2.jpg

    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
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  2. #2
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    Quote Originally Posted by chessmastery2001 View Post
    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.

    Click image for larger version. 

Name:	calculus.JPG 
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ID:	8984

    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.
    [IMG]file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/moz-screenshot.jpg[/IMG]
    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.

    Click image for larger version. 

Name:	calculus2.jpg 
Views:	363 
Size:	7.9 KB 
ID:	8985

    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
    1. Area B = \int_{0}^{1} x^2 \, dx = \, ....

    Area A = 1 - Area B.

    Use this as a model for how to answer the other questions.
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  3. #3
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    Calculus

    I need some more help with problems 2-4. I've attached a pdf file. HL IA Investigating Ratios of Areas and Volumes.pdf

    Thanks Again!
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  4. #4
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    Quote Originally Posted by chessmastery2001 View Post
    I need some more help with problems 2-4. I've attached a pdf file. HL IA Investigating Ratios of Areas and Volumes.pdf

    Thanks Again!
    If you understand how to do Q1 you should be able to attempt Q2-4. The pdf file looks like an assignment which will be assessed and contribute towards your final grade. Sorry, but help cannot be given under those circumstances.

    Thread closed.
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