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Math Help - double integral transformation using jacobian and green theorem

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    double integral transformation using jacobian and green theorem

    hello, can anyone gv me more ideas as to how to sketch the region over which the attched integral is to be evaluated and the transforming region. I have read through my note several times but still don't get it.
    I would be grateful if the second question can also be checked.

    Find attached the question.
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    Quote Originally Posted by dareken2001 View Post
    hello, can anyone gv me more ideas as to how to sketch the region over which the attched integral is to be evaluated and the transforming region. I have read through my note several times but still don't get it.
    I would be grateful if the second question can also be checked.

    Find attached the question.
    In Q1, the boundaries becomes the following:
    y = x becomes v = u,
    y = 2 becomes v = \frac{2}{u+1}
    x = 0 becomes either u = 0 or v = -1 but since v > 0 its the first choice.
    These three curves in the (u,v) plane enclosed a closed region.

    Q2. Using Greens theorem in the plane
    \int P dx + Q dy = \int \int Qx - Py dA
    If we choose Qx = y^2 and Py = -x^2 so that
    P = - x^2 y and Q = x y^2 and parameterize the ellipse as
    x = a \cos t and y = b \sin t then
    \int P dx + Q dy = \int_0 ^{2\pi} (a^3 b + a b^3)\sin^2t \cos^2t dt from which the result follows.

    Hope this helps
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