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    Hi, was wondering if anyone could help me with the following question:

    By changing varaibles u = xy and v = y/x, show that the integral (can be seen in my solution sheet) = 15/8 where R is the region in the first quadrant bounded by xy = 1, xy = 2, y = x and y = 2x.

    This is my solution so far:

    My Solution here

    I'm trying to follow the examples in my notes but I still don't understand the use of Jacobian? Can anyone help me with the question, thanks! :confused:
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    (Original post by rockiee)
    Hi, was wondering if anyone could help me with the following question:

    By changing varaibles u = xy and v = y/x, show that the integral (can be seen in my solution sheet) = 15/8 where R is the region in the first quadrant bounded by xy = 1, xy = 2, y = x and y = 2x.


    I'm trying to follow the examples in my notes but I still don't understand the use of Jacobian? Can anyone help me with the question, thanks! :confused:
    That region of the plane is given by 1 < u < 2 and 1 < v < 2. It's, from the uv-point of view, a rectangle but is in the xy-plane a stretched rectangle. The Jacobian (or rather its modulus) is a measure at each point of how much the area's getting stretched, which is why it goes into the double integral formula.

    Specifically you need to:
    write your integrand in terms of u and v
    change the limits to 1,2 and 1,2
    recall that dx dy = |J| du dv for the Jacobian you calculated.
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    Oh thanks!

    Hmm that's the part which I don't understand. Changing the integrand in terms of u and v? The example I have in my notes is so straightforward because u = the term in the integrand so I just substitute it in. However in this one I'm not sure how to approach the conversion of the integrand :confused: Can offer some help?

    so far I've got the integrand:

    ∫∫ (x²+2y²) | ∂(x,y)/∂(u,v) du dv
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    x = rt(u/v), y = rt(uv)
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    thanks!

    hmm I don't think I've integrated it properly....I got logs in my integration..is that right? :confused: and I didnt get 15/8 either
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    Your Jacobian is wrong - you've not got v_x correct
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    ohh its suppose to be -y/x² not y right? so the Jacobian is...1/(2y/x) = x/2y?
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    (Original post by rockiee)
    ohh its suppose to be -y/x² not y right? so the Jacobian is...1/(2y/x) = x/2y?
    which is 1/(2v)
 
 
 
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