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Double Integrals with Polar Coordinates

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

    Use polar coordinates to find the volume of the solid bounded by the paraboloid z = 47 - 5x2 - 5y2 and the plane z = 2.

    2. Relevant equations

    x2 + y2 = r2
    x = rcosθ
    y = rsinθ

    3. The attempt at a solution

    I substituted the z = 2 into the equation given,

    2 = 47 - 5x2 - 5y2
    45 = 5x2 + 5y2
    9 = x2 + y2

    So from here, I know that r = 3, and 0<r<3.
    Since it's a circle, I know that 0<θ<2∏

    Then, I know that x2 + y2 = r2, so,

    z = 47 - 5(x2+y2)
    = 47 - 5(r2)

    ∫[0,2∏]∫[0,3] (47 - 5(r2))rdrdθ

    When I take the double integral, I get (441∏)/2. This is incorrect. It seems like a very simple question, and my math looks correct. Have I made a conceptual mistake somewhere?

    Thanks for any help! :)
  2. jcsd
  3. Jul 4, 2013 #2


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    Staff: Mentor

    The height of the solid object is not the function value, as your lower border is z=2 instead of z=0.
  4. Jul 5, 2013 #3
    Ohhh. Yes, that makes sense. Thank you so much! :)
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