$(10$ points$)$ Let $D$ be the shaded region below $($the area inside the circle of radius $1$ centered at $(1,0),$ and outside the circle of radius $\frac{1}{2}$ centered at $(\frac{1}{2},0)).$

$($a$)(5$ points$)$ Express ${}_{D}f(x,y)dA$ as the sum of three double integrals in Cartesian coordinates, each in the order $dydx.$

$($b$)(5$ points$)$ Express ${}_D\sqrt[\phantom{2}]{x^2+y^2}dA$ in polar coordinates. Do not evaluate the integral.