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Use the given transformation to evaluate the integral

  1. Jul 17, 2012 #1
    ∫∫10xy(dA), where R is the region in the first quadrant bounded by the lines y=x/2 and y=2x/3 and by the hyperbolas xy=1/2 and xy=3/2

    The transformations given in the problem (these cannot be altered): x=u/v and y=v



    Relevant equations:

    The Jacobian - ∂(x,y)/∂(u,v)



    The attempt at a solution:

    y=(3/2)x: 2v^2=3u
    y=x/2: 2v^2=u
    xy=1/2: u=1/2
    xy=3/2: u=3/2

    After making the transformations, I get the following double integral:

    10∫(u from 1/2 to 3/2)∫(v from √(u/2) to √(3u/2)) (u/v)dvdu

    I feel like my transformations are suppose to result in basic bounds without variables, so I think I am doing this problem incorrectly and can't figure out any other way to go about it.
     
  2. jcsd
  3. Jul 17, 2012 #2

    SammyS

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    That looks good to me.

    The transformation does simplify the region, even if it's not rectangular.

    Try the integration.
     
  4. Jul 17, 2012 #3
    Wonderful! I performed the integration and got 10ln(√3) and this was correct.
     
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