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Homework Help: Change of Variables for Double Integral

  1. Aug 10, 2011 #1
    Update: I figured out how to solve the problem. Nevermind.

    1. The problem statement, all variables and given/known data

    Use a suitable change of variables to evaluate the double integral:


    2. Relevant equations

    [itex]\frac{\partial(x,y)}{\partial(u,v)}=( \frac{\partial x}{\partial u} )(\frac{\partial y}{\partial v})-(\frac{\partial x}{\partial v})(\frac{\partial y}{\partial u})[/itex]


    3. The attempt at a solution

    I made the following change of variables:



    (At first I tried u=y-2x and v=x+y but that didn't work)

    Next, I solved for x and y:



    I found that the Jacobian is:


    And therefore...


    Next I changed the boundaries:

    [itex]x=0\rightarrow v=0[/itex]

    [itex]x=1\rightarrow v=2[/itex]

    [itex]y=3-x\rightarrow u=3[/itex]

    [itex]y=2x\rightarrow u=\frac{3}{2}v[/itex]

    Therefore, the new region S on the uv-plane is the triangle described by:

    [itex]0\leq v\leq 2[/itex]

    [itex]\frac{3}{2}v\leq u\leq 3[/itex]

    The original integrand becomes:

    [itex](y-2x)e^{(x+y)^{3}}\rightarrow \frac{2u-3v}{2}e^{u^{3}}[/itex]

    So the new integral is:


    I used wolfram alpha to find out that the integral is equal to [itex]\frac{1}{18}(e^{27}-1)[/itex], which is the correct answer, but I am not sure how to evaluate the integral. Do I have to use another change of variables? Is there a better change of variables I could have used in the beginning?

    Thank you. Any help is much appreciated. :smile:


    Okay, I figured out how to solve the integral. I'll post the rest of the solution in case anybody else who has this same question ever happens across this post. What I did was I switched the order of integration so that the integral becomes:




    Use substitution with [itex]w=u^{3}[/itex] and [itex]dw=3u^{2}du\rightarrow \frac{1}{3}dw=u^{2}du[/itex]:



    Last edited: Aug 10, 2011
  2. jcsd
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