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Double integral in polar coordination

  1. Sep 26, 2010 #1
    1. The problem statement, all variables and given/known data


    2. Relevant equations

    3. The attempt at a solution

    Please tell me if I am wrong. I suspect about the ranges. Are my range corrrect?

    Attached Files:

    Last edited: Sep 26, 2010
  2. jcsd
  3. Sep 26, 2010 #2
    If you have a double integral on the form I = [tex]\int \int_{R} f(x,y) dy dx[/tex]
    Then if you to integrate using polar coordinates then it changes to
    I = [tex]\int \int_{R} f(x,y) dA[/tex] where

    [tex]dA = r dr d\theta[/tex]

    trying adding this and see if you don't get the right result.
  4. Sep 26, 2010 #3
    Dear Susanne217,

    Thank you very much for your reply, but i have already added [tex]dA = r dr d\theta[/tex]
    to get the result. If you calculate it you will see that I had already added it! But my problem is the ranges. I suspect about the ranges.
  5. Sep 26, 2010 #4
    The quantity you integrate (that is '1') is always positive.
    An integral is basically a sum, so a sum of positive terms cannot turn into something negative.
    The result of the book [tex](1-\sqrt2 \cong -0.41) [/tex] is negative, so it is wrong.

    [tex]=\int_0^{\pi/4} \int_{0}^{\frac{sen\theta}{cos^2\theta}} dr\: d\theta [/tex]

    [tex]=\int_0^{\pi/4} {\frac{sen\theta}{cos^2\theta}} \: d\theta [/tex]

    [tex]=\left[{\frac{1}{cos\theta}\right]_0^{\pi/4} [/tex]

    [tex]=\sqrt2-1 [/tex]

    Your result is ok.
    Last edited: Sep 26, 2010
  6. Sep 26, 2010 #5
    Thank you very much for the notification about "positive". But would you please tell me what your idea is about the ranges? especially about the ranges of "r". Are you sure that my ranges are correct?
  7. Sep 26, 2010 #6
    Yes they are correct because in a polar-cartesian system:
    the system
    [tex]y = x tg\theta[/tex]
    [tex]y = x^2[/tex]

    gives [tex]x = tg\theta[/tex]

    [tex]r = \sqrt(x^2+y^2)= tg\theta \sqrt(tg^2\theta+1) = tg\theta / cos\theta = sen\theta / cos^2\theta[/tex]
  8. Sep 26, 2010 #7
    Dear Quinzio,

    Thank you very much for your help.
  9. Sep 26, 2010 #8

    Attached Files:

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