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Thread: Substitutions in multiple integrals

  1. #1
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    Substitutions in multiple integrals

    R is a region in the first quadrant of the xy-plane bounded by xy=1, xy=9, y=x, and y=3x. Using the transformation $\displaystyle v=\sqrt{\frac{y}{x}}$, $\displaystyle u=\sqrt{xy}$ with u>0, v>0 to rewrite the integral below over an appropriate region G in the uv-plane. Then evaluate the uv-integral over G.

    $\displaystyle \displaystyle \int\int_{R}(\sqrt{\frac{y}{x}}+\sqrt{xy})dxdy$

    I'm totally lost on this. This is an exam review question but for all of our examples in the textbook the transformation is written in terms of x=... and y=...

    I have no idea what to do here.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by downthesun01 View Post
    R is a region in the first quadrant of the xy-plane bounded by xy=1, xy=9, y=x, and y=3x. Using the transformation $\displaystyle v=\sqrt{\frac{y}{x}}$, $\displaystyle u=\sqrt{xy}$ with u>0, v>0 to rewrite the integral below over an appropriate region G in the uv-plane. Then evaluate the uv-integral over G.

    $\displaystyle \displaystyle \int\int_{R}(\sqrt{\frac{y}{x}}+\sqrt{xy})dxdy$

    I'm totally lost on this. This is an exam review question but for all of our examples in the textbook the transformation is written in terms of x=... and y=...

    I have no idea what to do here.
    Observe that $\displaystyle xy=1\implies \sqrt{xy}=1$ and $\displaystyle xy=9\implies \sqrt{xy}=3$ (we can't allow for -3 given the restriction on u). Thus, we see that $\displaystyle 1\leq u\leq 3$.

    Similarly, we see that $\displaystyle y=x\implies \sqrt{\dfrac{y}{x}}=1$ and $\displaystyle y=3x\implies \sqrt{\dfrac{y}{x}}=\sqrt{3}$ (again, we can't allow for $\displaystyle -\sqrt{3}$ due to the restriction on v). Thus we see that $\displaystyle 1\leq v\leq \sqrt{3}$. We now have the limits for the transformed integral.

    Next, we need to get x and y in terms of u and v and then find its Jacobian.

    Observe that

    (1) $\displaystyle xv^2=y$ from the definition of v, and
    (2) $\displaystyle u^2=xy$ from the definition of u.

    Substitute (1) into (2) to see that $\displaystyle u^2=x^2v^2\implies x=\dfrac{u}{v}$.

    Multiply the corresponding sides of (1) and (2) together to get $\displaystyle xu^2v^2=xy^2\implies y=uv$

    Now find the Jacobian. I leave it for you to verify that $\displaystyle J(u,v)=\dfrac{2u}{v}$.

    Therefore, $\displaystyle \displaystyle\iint\limits_R\left(\sqrt{\dfrac{y}{x }}+\sqrt{xy}\right)\,dx\,dy=\int_1^{\sqrt{3}}\int_ 1^3\left(2u+\frac{2u^2}{v}\right)\,du\,dv$

    Can you proceed? Does this all make sense?
    Last edited by Chris L T521; Nov 22nd 2010 at 10:09 PM.
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    You helped tons. Thanks a bunch.
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