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Change of variables cylindrical coordinates

  1. May 13, 2013 #1
    1. The problem statement, all variables and given/known data
    Let S be the part of the cylinder of radius 9 centered about z-axis and bounded
    by y >= 0; z = -17; z = 17. Evaluate
    [itex]\iint xy^2z^2[/itex]


    2. Relevant equations



    3. The attempt at a solution
    So I use the equation [itex]x^2 + y^2 \leq 9[/itex], meaning that r goes from 0 to 3
    Since [itex]y \geq 0[/itex], θ goes from 0 to ∏
    So the integral looks like this:
    [itex]\int_0^∏ \int_0^3 \int_{-17}^{17} (rcosθ)(rsinθ)^2 z^2 r dzdrdθ[/itex]
    And I get:
    [itex]\int_0^∏ \int_0^3 \int_{-17}^{17} r^4 cosθsin^2θ z^2 dzdrdθ[/itex]
    I'm having trouble evaluating this integral because for the [itex]\int_0^∏ cosθsin^2θ [/itex]part. I get 0 (When u=sinθ by u-substitution and you get [itex](1/3)sin^3θ [/itex] from 0 to ∏)
    Basically I would like to know if my limits and my setup is correct, and if anyone can help me out with a solution I would be grateful.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 13, 2013 #2
    Well, firstly, the range of the dr-integral should be 0 to 9 not 0 to 9 because we're dealing with the cylinder of radius 9, not the equation x^2+y^2 <= 9.

    But that won't affect the problem, I just did this myself and I also got 0. Unless someone else can point out a mistake we both made, I think that's the answer.

    Edit: That makes sense, I think, because the equation w=xy^2z^2 is symmetric across the y axis.
     
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