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Math Help - Change of Variables in Multiple Integrals

  1. #1
    Super Member angel.white's Avatar
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    Change of Variables in Multiple Integrals

    Having a really tough time figuring out how to set this problem up:

    Use the given transformation to evaluate the integral.

    \int \int_R x^2 ~dA, where R is the region bounded by the ellipse 9x^2+4y^2 = 36;

    x=2u

    y=3v
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  2. #2
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    Quote Originally Posted by angel.white View Post
    Having a really tough time figuring out how to set this problem up:

    Use the given transformation to evaluate the integral.

    \int \int_R x^2 ~dA, where R is the region bounded by the ellipse 9x^2+4y^2 = 36;

    x=2u

    y=3v
    dA = dx \, dy \rightarrow J(u, v) \, du \, dv = [(2)(3) - (0)(0)] \, du \, dv = 6 \, du \, dv.

    R_{xy}: ~  9 x^2 + 4 y^2 \leq 36 ~ \rightarrow R_{uv}: ~ 36 u^2 + 36 v^2 \leq 36 \Rightarrow u^2 + v^2 \leq 1.

    So the double integral becomes \int \int_{R_{uv}} (2u)^2 (6) \, du \, dv = 24 \int \int_{R_{uv}} u^2 \, du \, dv.

    To exploit the circular region in the uv-plane I'd suggest switching to polar coordinates now to seal the deal.
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