$(10$ points$)$ Convert ${\displaystyle \underset{-1}{\overset{1}{}}}{\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{1-y^2}}{}}}{\displaystyle \underset{x^2+y^2}{\overset{\sqrt[\phantom{2}]{x^2+y^2}}{}}}xyzdzdxdy$ into $an$ integral $in$ cylindrical coordinates $(4$

points$),$ and evaluate the integral $(6$ points$).$