Answer
$$\dfrac{195 \pi}{2}$$
Work Step by Step
Green's Theorem states that:
$\oint_C A\,dx+B \,dy=\iint_{D}(\dfrac{\partial B}{\partial x}-\dfrac{\partial A}{\partial y}) dA $
Conversion to polar coordinates is as follows: $x=r \cos \theta; y= r \sin \theta$
We need to set up the line integral and compute the integrand of the double integral as follows:
$$\oint_C [-3y^2+3x^2] dA=\iint_{D}(\dfrac{\partial (3x^2)}{\partial x}-\dfrac{\partial (-3y^2) }{\partial y})dA \\=3 \times \int_{0}^{2 \pi} \int_{2}^{3} r^2 \times r dr d \theta \\=3 \times \int_{0}^{2 \pi} \int_{2}^{3} r^3 dr d \theta \\=3 \times \int_{0}^{2 \pi} [r^4/4]_{2}^{3} d \theta \\=-3 \times \int_0^{2 \pi} [\dfrac{r^4}{4}]_2^3 d \theta \\=\dfrac{195 \pi}{2}$$
