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Double integral problem

  1. Dec 8, 2011 #1
    1. The problem statement, all variables and given/known data

    Evaluate: ∫1 to 4∫0 to y(2/(x^2+y^2))dxdy

    2. Relevant equations



    3. The attempt at a solution

    So I know you have to spilt it up and do the dx integral first:

    ∫0-y(2/(x^2+y^2))dx

    Now this is where I don't know if I'm doing it right, I moved the 2 outside the integral and split up the fraction, so:

    2(∫1/x^2dx+∫1/y^2dx)

    Now since I'm only dealing with dx I'll ignore the y for right now:

    ∫1/x^2= -1/x|0to y
    = -1/y

    So the new integral is:

    ∫-2/(y+y^2)dy

    Again move the two outside and split up the intgeral:

    -2(∫1/ydy-∫1/y^2dy)
    -2(lny+1/y^2)from 1 to 4

    then it's just imputing numbers.

    So basically if you could tell me if I'm right about being able to split up the fraction like I do I'd very much appreciate it!
     
  2. jcsd
  3. Dec 8, 2011 #2

    SammyS

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    [itex]\displaystyle\frac{1}{x^2+y^2}\ne\frac{1}{x^2}+ \frac{1}{y^2}[/itex]

    Treat y as a constant when integrating with respect to x.

    [itex]\displaystyle \int\frac{1}{x^2+a^2}\,dx=\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+C[/itex]
     
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