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Math Help - Volume of solid

  1. #1
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    Volume of solid

    Find the volume of a solid bounded by the paraboloid z = 4 x2 y2 and the xy plane.
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  2. #2
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    Quote Originally Posted by winsome View Post
    Find the volume of a solid bounded by the paraboloid z = 4 – x2 – y2 and the xy plane.
    Have you drawn the region and determined your bounds for integration?
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  3. #3
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    Yes i have solved it as below but not sure that i m correct and now stuck to get it complete solution. Please help me to get its complete solution


    from to given equation 4 x2 y2 = 0

    Solve for y i.e.

    y2 = 4 x2
    y =

    Also for y = 0,

    4 x2 = 0
    x = 2

    Hence, Volume is given by

    V =

    Let substitute,

    x2 + y2 = r2 _______(2)

    x = r sin q dx = r cos q dq

    y = r cos q dy = r sin q dq
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  4. #4
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    Quote Originally Posted by winsome View Post
    Yes i have solved it as below but not sure that i m correct and now stuck to get it complete solution. Please help me to get its complete solution


    from to given equation 4 x2 y2 = 0

    Solve for y i.e.

    y2 = 4 x2
    y =

    Also for y = 0,

    4 x2 = 0
    x = 2

    Hence, Volume is given by

    V =

    Let substitute,

    x2 + y2 = r2 _______(2)

    x = r sin q dx = r cos q dq

    y = r cos q dy = r sin q dq
    \displaystyle V = \int \int_{R_{xy}} 4 - x^2 - y^2 \, dy \, dx where R_{xy} is the region in the xy-plane defined by the interior of the circle x^2 + y^2 = 4.

    Switch to polar coordinates:

    \displaystyle V = \int_{0}^{2 \pi} \int_0^2 (4 - r^2) r \, dr \, d\theta = ....
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  5. #5
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    Quote Originally Posted by mr fantastic View Post
    \displaystyle V = \int \int_{R_{xy}} 4 - x^2 - y^2 \, dy \, dx where R_{xy} is the region in the xy-plane defined by the interior of the circle x^2 + y^2 = 4.

    Switch to polar coordinates:

    \displaystyle V = \int_{0}^{2 \pi} \int_0^2 (4 - r^2) r \, dr \, d\theta = ....

    how to get limits 0 to 2pi and 0 to 2
    Last edited by winsome; February 11th 2011 at 01:49 AM.
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  6. #6
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    Quote Originally Posted by winsome View Post
    i m unable to show my next steps with integration sign, so please u show me its next steps after itegration between 0 to 2 pi and 0 to 2.
    Integrate with respect to r. If you cannot do this then:

    1. I don't know how you got to the stage of studying multivariable calculus.

    2. I suggest you go back and review basic integration techniques.

    Please show you work and say where you get stuck.
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  7. #7
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    Quote Originally Posted by mr fantastic View Post
    Integrate with respect to r. If you cannot do this then:

    1. I don't know how you got to the stage of studying multivariable calculus.

    2. I suggest you go back and review basic integration techniques.

    Please show you work and say where you get stuck.
    actually i want to know how to convert limits -2 to +2 and -sq root(4 - x^2) to +sq root(4 - x^2) in 0 to 2pi and 0 to 2 in our integration
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  8. #8
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    Quote Originally Posted by winsome View Post
    how to get limits 0 to 2pi and 0 to 2
    Draw the circle! (Have you met polar coordinates before?)
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