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Integral ? please help

  1. Nov 7, 2009 #1
    1. The problem statement, all variables and given/known data
    double integral of 2/(2-x^2+y^2) x's from -y to y and y's from 0 to sqrt(2)/2
    3. The attempt at a solution
    okay so i first started by using a trig substitution
    and can i call a=(2+y^2) my a to simplify things so i get
    2/(a-x^2) x=sqrt(a)sin(t)
    dx=sqrt(a)cos(t)dt
    then we get 2/(sqrt(a)) ln|sec(t)+tan(t)| evaluated from -y to y
    then i get 2ln|(y+sqrt(a))(sqrt(a)-y))| after i simplified then I’m not sure what to do here .
     
  2. jcsd
  3. Nov 7, 2009 #2

    lanedance

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    have you considered a double variable change?

    based on the shape of the intergation region, i looked at a basis rotated by pi/4 & scaled to simplifythe integral, which ithink simplified things a fair bit...

    that said it still a bit messy & i didn't follow it all the way through...
     
    Last edited: Nov 7, 2009
  4. Nov 7, 2009 #3
    this original integral was rotated by pi/4 , do you know where i could look at an example of this.
     
  5. Nov 7, 2009 #4

    lanedance

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    do you man you had already rotated teh coordinates?

    not really sure about an example, but by the rotation i meant try a subsititution something like
    y = u+v
    x = u-v

    this give the roatated basis frame. You just need to compute a Jacobian (which will eb a constat number in thsi case) & work out the limits

    for the region of integration, i had it as a (90,45,45) triangle with hypotenuse at y = 1/sqrt(2) and short sides along y=x, and y=-x

    in the variable change, the y=x and y=-x beome the varibale axis (u&v) and with a bit of work you should be able to read off the limts

    though as mentioned i haven't tried this through fully, just seems like its worth an attempt
     
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