$\int_0^{2\pi}\mathrm{d}\varphi\int_0^2(1-2\rho \cos \varphi-3\rho \sin \varphi)\rho\,\mathrm{d}\rho=\int_0^{2\pi} \left[\frac{\rho^2}{\color{red}2}-2\frac{\rho^3}{3} \cos \varphi-3\frac{\rho^3}{3} \sin \varphi \right]_0^2\mathrm{d}\varphi$
and the result is $4\pi$.