Let $D$ be the region between the circles $C_1$ and $C_2$ centered at the origin of radius $3$ and $4,$ respectively. Assume $C_1$ and $C_2$ are oriented positively with respect to $D.$ Use the extended version of Green's theorem for circulation to calculate

$o{\displaystyle \underset{c_1}{\overset{\phantom{}}{}}}\frac{-3x}{\sqrt[\phantom{2}]{x^2+y^2}}dx+\frac{4y}{\sqrt[\phantom{2}]{x^2+y^2}}dy+o{\displaystyle \underset{c_2}{\overset{\phantom{}}{}}}\frac{-3x}{\sqrt[\phantom{2}]{x^2+y^2}}dx+\frac{4y}{\sqrt[\phantom{2}]{x^2+y^2}}dy.$