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Let $R$ be the region in the first quadrant of the $x y$ -plane bounded

by the hyperbolas $x y=1, x y=9$ and the lines $y=x, y=4 x$ .

Use the transformation $x=u / v, y=u v$ with $u>0$ and $v>0$

to rewrite

$$\iint_{R}\left(\sqrt{\frac{y}{x}}+\sqrt{x y}\right) d x d y$$

as an integral over an appropriate region $G$ in the $u v$ -plane. Then

evaluate the $u v$ -integral over $G .$

See explanation for result.

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