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Math Help - Volume Integral

  1. #1
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    Volume Integral

    Question:
    The volume V between
    z = \frac{5{x}^{2} + 3{y}^{2}}{R}

    And the x, y plane (z = xy?)

    What is the volume of this region?

    Thoughts?
    \iiint dV = \iiint dx dy dz
    Given that this is a paraboloid (3D) im not too sure how to approach it. I guess it just wants the volume of the paraboloid not entirely sure if the z = xy changes anything seeming the paraboloid doesn't go into negative z anyway?

    Any help on how i should approach and deal with this is much appreciated!
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  2. #2
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    Re: Volume Integral

    Quote Originally Posted by imagemania View Post
    Question:
    The volume V between
    z = \frac{5{x}^{2} + 3{y}^{2}}{R}

    And the x, y plane (z = xy?)

    What is the volume of this region?

    Thoughts?
    \iiint dV = \iiint dx dy dz
    Given that this is a paraboloid (3D) im not too sure how to approach it. I guess it just wants the volume of the paraboloid not entirely sure if the z = xy changes anything seeming the paraboloid doesn't go into negative z anyway?

    Any help on how i should approach and deal with this is much appreciated!
    What is R?
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  3. #3
    Senior Member DeMath's Avatar
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    Re: Volume Integral

    Maybe z=x+y or z=x-y??
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  4. #4
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    Re: Volume Integral

    Sorry, R is a constant
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  5. #5
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    Re: Volume Integral

    The xy plane is the plane given by z = 0 .
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  6. #6
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    Re: Volume Integral

    Sorry im still not following,

    I understand teh diagram looks of the order of this:
    http://img202.imageshack.us/img202/4418/captureidmd.png

    And this questions asks for the volume between that and the xy plane i.e. the infinite square that supports it.

    Though i can't see converting the coordinate system as useful. I thought about doing:
    \vec{N} = r_{x} \times r_{y}
    Where:
    r_{x} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial x} ...
    but then N is a vector and not useful here as i want scalars!
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  7. #7
    Senior Member DeMath's Avatar
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    Re: Volume Integral

    Quote Originally Posted by imagemania View Post
    Question:
    The volume V between
    z = \frac{5{x}^{2} + 3{y}^{2}}{R}

    And the x, y plane

    What is the volume of this region?
    Any help on how i should approach and deal with this is much appreciated!
    Maybe the author of this problem implied such equations of planes: z=x,~z=y??

    Then we can calculate the finite volume.
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  8. #8
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    Re: Volume Integral

    I'll word the question fully, V will be the volume
    Consider V inside the cylinder
    x^{2} + y^{2} = 6{R}^{2}
    and between z = (\frac{5{x}^{2} + 3{y}^{2}}{R}) and the (x, y) plane.

    x, y and z are the Cartesian coordinates and R is also a constant.



    Now i read this question and thought it meant "Find the volume of the cylinder then find teh volume betwen the other two independently" i.e. two separate questions. Perhaps i was wrong?

    Finding V for teh cylinder is just a matter of finding the jacobian and using cylindrical coordinates surely?

    But the other part, assuming it is a separate question, i am not sure how to best approach it

    Thanks
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  9. #9
    Senior Member DeMath's Avatar
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    Re: Volume Integral

    The (x, y) plane given by z=0. So



    See the picture

    Last edited by DeMath; October 16th 2011 at 02:56 AM.
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  10. #10
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    Re: Volume Integral

    Oh i see now DeMath, it is simply using the first as z the and then introducing the cylindrical coordinates - didn't help i miss understood the question thanks a lot

    Though i must find out, how did you draw that image? I currently use Maple (14) and haven't figured how to draw them within the same body, what did you use and how did you do it?
    Last edited by imagemania; October 18th 2011 at 03:53 AM.
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  11. #11
    Senior Member DeMath's Avatar
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    Re: Volume Integral

    Quote Originally Posted by imagemania View Post
    Though i must find out, how did you draw that image? I currently use Maple (14) and haven't figured how to draw them within the same body, what did you use and how did you do it?
    You're lucky - this is done in Maple
    Here is the code of this body (for R=\sqrt{6})

    plot3d([(5x^2+3y^2)/sqrt(6)], x=-6..6, y=-sqrt(36-x^2)..sqrt(36-x^2), filled=true, style=hidden, color="Cyan", lightmodel=light2, transparency=0.25, numpoints=10000, axes=normal, orientation=[81, 60])

    When you see the picture, then move the cursor on her, click the right key of the mouse and after Esc.
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