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Math Help - finding volume of a solid using polar coordinates

  1. #1
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    finding volume of a solid using polar coordinates

    I love polar coordinates, but not very familiar with them...
    Here is the problem:

    Use polar coordinates to find the volume of the given solid.
    Below the paraboloid z = 36 - 9x^2 - 9y^2 and above the xy-plane.

    My attempt:

    the paraboloid can be rewritten as x^2+y^2 = 4
    in this case, i thought the limits in polar coordinates would be:
    0 <= theta <= pi
    0 <= r <= 2
    since the radius of the circle is 2 and is only in the first two quadrants.
    since r = x^2 + y^2.... it is just the double integral of r^2 dr d-theta. in that case i got my final answer to be 8/3(pi).

    Where did I get my reasoning wrong? Thanks
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  2. #2
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    Have you learned about cylindrical coordinates? In either case, you are correct that x^2+y^2=4 bounds the region, but it is only the circular boundary of the region, not the paraboloid itself. To find the volume using cylindrical coordinates, we calculate

    \int_0^2\int_0^{2\pi}\int_0^{36-9r^2}r\,dz\,dr\,d\theta.

    If you're not yet familiar with cylindrical coordinates, this just amounts to

    \int_0^2\int_0^{2\pi}(36-9r^2)\,r\,dr\,d\theta.

    Notice that we have included an extra factor r to make up for uneven coordinate distribution in polar and cylindrical coordinates.
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  3. #3
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    Thanks Scott,
    we havent learned cylindrical coordinates. This section of homework specifically calls for polar coordinates anyway. Thanks for the explanation
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