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Math Help - Integration Question

  1. #1
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    Integration Question

    I'm having a bit of trouble with this question:

    In what ratio does te X-axis divide the area of the region bounded by the parabolas y=4x-x^2 and y=x^2-x
    Last edited by acevipa; February 14th 2009 at 06:21 PM.
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  2. #2
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    Hello, acevipa!

    I assume you made a sketch . . .


    In what ratio does the x-axis divide the area of the region
    bounded by the parabolas: . y\:=\:4x-x^2\,\text{ and }\,y\:=\:x^2-x
    Code:
                |
           *    |          *
                |
                |
            *   |    *    *
                | *:::::*
             *  |*:::::::*
              * |:::::::*
        - - - - *:-:-:* - * - - - -
                |  *
                |
               *|          *
                |

    The parabola intersect when: . x^2-x \:=\:4x-x^2 \quad\Rightarrow\quad x \:=\:0,\:\tfrac{5}{2}


    Find the total area between the parabolas: . A_{\text{total}} \;=\;\int^{\frac{5}{2}}_0\bigg[(4x-x^2) - (x^2-x)\bigg]\,dx

    Find the area below the x-axis: . A_{\text{below}} \;=\;-\int^1_0\bigg[x^2-x\bigg]\,dx


    Find the area above the x-axis: . A_{\text{above}} \;=\;A_{\text{total}} - A_{\text{below}}


    Finally, form the ratio: . \frac{A_{\text{above}}}{A_{\text{below}}}

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