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U substitution and integration

  1. Oct 29, 2012 #1
    1. The problem statement, all variables and given/known data

    use the substitution u= x+y and v=y-2x to evaluate double integral from
    ∫1-0∫(1−x) -(0) of (√x+y) (y−2x)^2 dydx

    2. Relevant equations

    integration tables im assuming

    3. The attempt at a solution
    i tried to integrate directly but none of my integration tables match up to the format
     
  2. jcsd
  3. Oct 29, 2012 #2
    You should do as the assignment tells you to do.

    So perform the change of variables first.
     
  4. Oct 29, 2012 #3
    I will write it in latex for those who want to solve it:
    [tex]\int^{1}_{0} \int^{1-x}_{0} \sqrt{x+y} (y-2x)^2\,dy\,dx[/tex]
    Also, the Jacobian of the transformation you are trying to perform is
    [tex]\begin{vmatrix}
    1 & 1 \\
    -2 & 1
    \end{vmatrix}[/tex]
    What does that equal?
     
  5. Oct 29, 2012 #4
    the jacobian equals 3 but how is that related to the entire problem ? :s
     
  6. Oct 29, 2012 #5

    SammyS

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    You need the Jacobian to change dy dx to du dv or dv du .

    You will also need to change the limits of integration.

    Solving the system of equations,
    u= x+y

    v=y-2x​

    for x & y, will help you to do that.

    Sketch the region of integration for the given integral, [itex]\displaystyle \int^{1}_{0} \int^{1-x}_{0} \left(\sqrt{x+y\ }\, (y-2x)^2\right)\,dy\,dx\,,\ [/itex] in the xy-plane. Then convert that to the corresponding region in the uv-plane.
     
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