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Change of variables

  1. Jun 19, 2009 #1
    1. The problem statement, all variables and given/known data
    Let D be the region bounded by x=0, y=0, x+y=1, x+y=4. Using the change on variables x=u-uv, y=uv and the jacobian, evaluate the double integral
    double integral of dxdy/(x+y)

    2. Relevant equations
    answer is 3


    3. The attempt at a solution
    i drew the graph and found the boundaries x=1,y=0,x+y=1,x+y=4 and solved u and v for each boundary

    for x=0: 0=u-uv
    v=1

    for y=0: 0=uv
    so either v=0 or v=0 or both u and v are 0

    for x+y=1: u-uv+uv=1
    u=1

    for x+y=4: u-uv+uv=4
    u=4

    therefore i have the region in the uv plane [1,4]X[0,1]

    now dx=(1-v)du
    dy=udv

    therefore i get double integral(i-v)dvdu dudv with v varying from 0 to1 and u varying from 1 to 4 and i get 3/2?

    im not sure im doin the right method, i have a feeling im wrong when i let y=0 and assume v=0?
     
    Last edited: Jun 19, 2009
  2. jcsd
  3. Jun 19, 2009 #2
    ok i forgot bout the jacobian so i tried J=
    |delx/delu delx/delv |
    |dely/delu dely/delv |
    and i got J=u, which gives the right answer, yet im not sure if im doin this right i feel i fluked the answer
     
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