Consider the integral ${\displaystyle \underset{0}{\overset{1}{}}}{\displaystyle \underset{-\sqrt[\phantom{2}]{2-y^2}}{\overset{-y}{}}}{x}^2{y}^{2}dxdy.$

$($a$)(2$ Points$)$ Sketch the region of integration.

$($b$)(4$ Points$)$ Evaluate by changing to polar coordinates.