= Solution
<Green theorem> states that for a positively oriented planar region $D$,
$$
\oint_{\partial D}P\,dx+Q\,dy
=\iint_D\left(\frac{\partial Q}{\partial x}
-\frac{\partial P}{\partial y}\right)dA.
$$
Here $P=xy^2-\lambda y$ and $Q=x^2y$, so
$$
Q_x-P_y=2xy-(2xy-\lambda)=\lambda.
$$
The stated orientations make $C_1\cup C_2$ the positive boundary of the region inside the radius-$a$ circle and outside the radius-$b$ circle. Its area is $\pi(a^2-b^2)$ regardless of the inner circle's center. Therefore
$$
\boxed{\int_{C_1\cup C_2}P\,dx+Q\,dy
=\pi\lambda(a^2-b^2)}.
$$
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