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Homework Help: Derivation: Area of Ellipse

  1. Jul 16, 2008 #1
    1. The problem statement, all variables and given/known data
    Show that the area of [tex]x^2/a^2+y^2/b^2=1[/tex] is [tex]\pi ab[/tex]

    2. Relevant equations
    Given transformations:
    [tex]x=au[/tex]
    [tex]y=bv [/tex]

    3. The attempt at a solution

    [tex]J(u,v) = a*b [/tex]

    [tex] \int\int ((au)/a)^2+((bv)/b)^2 J(u,v) dudv [/tex]

    [tex]\int\int u^2+v^2 J(u,v) dudv [/tex]

    [tex]\int_0^{2\pi}\int_0^1 r^2 J(u,v) r drd\theta [/tex]

    [tex]\int_0^{2\pi}\int_0^1 a b r^3 drd\theta [/tex]

    [tex]\int_0^{2\pi} 1/4 a b d\theta [/tex]

    =[tex]\frac{\pi a b}{2}[/tex]

    But that's obviously wrong. Where did I mess up?
     
  2. jcsd
  3. Jul 16, 2008 #2

    arildno

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    Right from the start lies your mistake!
    You are to integrate:
    [tex]\int_{A}dA=\int_{A}dxdy=\int_{A}abdudv=\int_{0}^{2\pi}\int_{0}^{1}abrdrd\theta=\pi{ab}[/tex]
     
  4. Jul 16, 2008 #3
    :eek:

    I can't believe I even had to ask this! Thanks so much!!!
     
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