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Help with Changing of variables, Jacobian, Double Integrals?

  1. Mar 7, 2012 #1

    Suy

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

    Show that T(u,v) = (u2 - v2, 2uv)
    maps to the triangle = {(u,v): 0 ≤ v ≤ u ≤ 4} to the domain D
    bounded by x=0, y=0, and y2 = 1024 - 64x.

    Use T to evaluate ∬D sqrt(x2+y2) dxdy

    2. Relevant equations

    3. The attempt at a solution

    x=u2-v2
    y=2uv
    Jacobian= 4u2+4v2 dudv
    I guess the equation in the changed variable integral should be ∫∫sqrt((u2-v2)2+(2uv)2) (4u2+4v2) dudv

    But, I don't know how to get the bounds for the integrals in terms of u and v.
    Can someone help me on this??
     
  2. jcsd
  3. Mar 7, 2012 #2
    The first part of the problem already told you what the bounds of u and v are
     
  4. Mar 7, 2012 #3

    Suy

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    I know that 0 ≤ u ≤ 4 and 0 ≤ v ≤4.
    For y^2 = 1024 - 64x, x is restricted from 0 ≤ x ≤16 and 0 ≤ y ≤32??
     
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