Let $D$ be the solid bounded above by the sphere centered at origin with radius $\sqrt[\phantom{2}]{3}$ and below by the plane $z=1.$ Which of the following represents the integral ${}_D{x}^{2}zdzdydx?$

$($a$),{\displaystyle \underset{-1}{\overset{1}{}}}{\displaystyle \underset{-\sqrt[\phantom{2}]{1-x^2}}{\overset{\sqrt[\phantom{2}]{1-x^2}}{}}}{\displaystyle \underset{2}{\overset{\sqrt[\phantom{2}]{2-x^2-y^2}}{}}}{x}^{2}zdzdydx$

$($b$),{\displaystyle \underset{-1}{\overset{1}{}}}{\displaystyle \underset{-\sqrt[\phantom{2}]{2-x^2}}{\overset{\sqrt[\phantom{2}]{2-x^2}}{}}}{\displaystyle \underset{1}{\overset{\sqrt[\phantom{2}]{3-x^2-y^2}}{}}}{x}^{2}zdzdydx$

$($c$),{\displaystyle \underset{-\sqrt[\phantom{2}]{2}}{\overset{\sqrt[\phantom{2}]{2}}{}}}{\displaystyle \underset{-\sqrt[\phantom{2}]{1-x^2}}{\overset{\sqrt[\phantom{2}]{1-x^2}}{}}}{\displaystyle \underset{1}{\overset{\sqrt[\phantom{2}]{2-x^2-y^2}}{}}}{x}^{2}zdzdydx$

$($d$),{\displaystyle \underset{-2}{\overset{2}{}}}{\displaystyle \underset{-\sqrt[\phantom{2}]{2-x^2}}{\overset{\sqrt[\phantom{2}]{2-x^2}}{}}}{\displaystyle \underset{1}{\overset{\sqrt[\phantom{2}]{3-x^2-y^2}}{}}}{x}^{2}zdzdydx$

$($e$),{\displaystyle \underset{-\sqrt[\phantom{2}]{2}}{\overset{\sqrt[\phantom{2}]{2}}{}}}{\displaystyle \underset{-\sqrt[\phantom{2}]{2-x^2}}{\overset{\sqrt[\phantom{2}]{2-x^2}}{}}}{\displaystyle \underset{1}{\overset{\sqrt[\phantom{2}]{3-x^2-y^2}}{}}}{x}^{2}zdzdydx$