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Polar Double Integrals

  1. Dec 4, 2013 #1
    1. The problem statement, all variables and given/known data

    The plane z = 2 and the paraboloid z = 8 − 6x2 − 6y2 enclose a solid. Use polar coordinates to find the volume of this solid.

    2. Relevant equations

    ∫∫R f(x,y) dA = ∫βαba f(rcosθ, rsinθ) r dr dθ

    3. The attempt at a solution

    z = 2, z = 8 − 6x2 − 6y2

    Setting these two equal, we can find where the two functions intersect.

    2 = 8 − 6x2 − 6y2

    so

    0 = 6 − 6x2 − 6y2

    Solving for x and y, we get

    1 = x2 + y2

    So the intersection is a circle of radius 1 on the plane z = 2.

    Knowing this, we can write the domain of x and y both in terms of r and θ:

    { r,θ | 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2∏}

    Using this domain, I set up my double integral with the above layout found in my provided equations section. I don't feel like typing all the integrations out, but is my above process wrong? If not, I can focus on finding errors in my integration and ask further questions as needed.

    Thanks.
     
    Last edited: Dec 4, 2013
  2. jcsd
  3. Dec 5, 2013 #2

    haruspex

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    Looks to me that you are in danger of including the volume below z=2 (within the circle).
     
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