Using Green's Theorem, compute the counterclockwise circulation $I$ of $\vec{F}=\langle-\sqrt{x^2+y^2},\sqrt{x^2+y^2}\rangle$ around the region defined by the polar coordinate inequalities $7 \leq r \leq 8$ and $0 \leq \theta \leq \pi$.
My inclination is to approach as follows: $$ \\ \displaystyle \\ Q = \sqrt{x^2+y^2} \implies Q_x = \dfrac{ x}{\sqrt{x^2+y^2}} \\ P = -\sqrt{x^2+y^2} \implies P_y = \dfrac{-y}{\sqrt{x^2+y^2}} \\ \\ \\ x = r\cos{\theta} \\ y = r\sin{\theta} \\ \implies \vec{F} = \langle -r, r \rangle \\ \implies I = \int\int_R \left(Q_x-P_y\right) \mathrm{d}A = \int_0^\pi \int_7^8 \frac{r^2(\cos{\theta}+\sin{\theta})}{r}\mathrm{d}r\mathrm{d}\theta \\ \implies I = \int_0^\pi (\cos{\theta} + \sin{\theta}) \ \mathrm{d}\theta \cdot \int_7^8 r \ \mathrm{d}r $$ Evaluate and find $I = -15$.
Is this effective, valid, both, or neither?