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Change of variables in integration.

  1. Mar 11, 2012 #1
    1. The problem statement, all variables and given/known data

    The original integral is
    $$\left[\int_0^{\infty} {\int_0^{\infty} {F(x + y,x - y) \cdot dx \cdot dy} } \right]$$

    What should be the limits of the integrals. (position represented by '?' symbol)
    $$\left[\int_?^? {\int_?^? {F(u,v) \cdot (\frac{1}{2})du \cdot dv} } \right]$$ .

    When x+y is substituted by 'u' and x-y is substituted by 'v'
    2. Relevant equations

    use x+y= u , x-y=v, I am confused about the limits of the integrals

    3. The attempt at a solution
    dx*dy = 0.5 * du * dv - This I got by using Jacobian matrix.
    I need help in deciding the limits of the integrals.

    My approach:
    $$\left[\int_0^{\infty} {\int_{\left| v \right|}^{\infty} {F(u,v) \cdot (\frac{1}{2})du \cdot dv} } \right]$$

    Is this correct???
    Last edited by a moderator: Mar 11, 2012
  2. jcsd
  3. Mar 11, 2012 #2


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    Science Advisor
    Homework Helper

    hi vineel49! :smile:
    try drawing it …

    that's simply rotating the axes by 45°, isn't it? :wink:

    (and scaling up or down by the Jacobian)

    so if one of the new limits is 0 to ∞, the other must be … (not 0) to … ? :smile:
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