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Math Help - Calculating area using Integrals

  1. #1
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    Calculating area using Integrals

    calculate the surface of the area determined with curve 81y^2=4x^2(9-x^2)
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  2. #2
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    Quote Originally Posted by the prince View Post
    calculate the surface of the area determined with curve 81y^2=4x^2(9-x^2)
    your problem statement is incomplete. Is this enclosed region rotated about an axis? ... or are you simply wanting the area enclosed by the lemniscate?
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  3. #3
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    Hello, the prince!

    I assume you want the area of the region . . .


    Calculate the area of the region determined by: . 81y^2\:=\:4x^2(9-x^2)

    We have: . y \;=\;\pm\frac{2}{9}x\sqrt{9-x^2}

    The graph is symmetric to the x-axis, y-axis and the origin
    . . and has intercepts: . (0,0),\;(\pm3,0)

    The graph looks like a infinty-sign.
    Code:
                      |
           * *        |        * *
       *         *    |    *         *
                    * | *
    - * - - - - - - - * - - - - - - - * -
                    * | *
       *         *    |    *         *
           * *        |        * *
                      |

    As skeeter pointed out, it is called a lemniscate.


    The total area is: . A \;=\;4 \times \tfrac{2}{9}\int^3_0 x\left(9-x^2\right)^{\frac{1}{2}}dx

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