(15 pt) (a) Consider the new coordinates $x = \frac{u}{v}$, $y = uv$ ($u > 0$, $v > 0$) to evaluate the integral
$\iint_R \left(\sqrt{\frac{y}{x}} + \sqrt{xy}\right) dx\ dy$,
where $R$ is the region in the first quadrant bounded by the hyperbolas $xy = 1$, $xy = 9$ and the lines
$y = x$, $y = 4x$.
(15 pt) (b) Let $C$ be the boundary of the region $R$ enclosed by the curves $(x - 1)^2 + y^2 = 1$ and $y^2 = x$.
Find the line integral $I = \oint_C (x^2 - y^2) dx + x^2 dy$ with limits of integration (do not calculate the integral,
do not use Green's theorem).
