$($Sections $3\mathrm{.}2,13\mathrm{.}3,13\mathrm{.}7,14\mathrm{.}3,14\mathrm{.}6,14\mathrm{.}7)$

Consider the integral

${\displaystyle \underset{-1}{\overset{1}{}}}{\displaystyle \underset{0}{\overset{\sqrt[\phantom{2}]{1-x^2}}{}}}1-y^{2}dydx$

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

$($b$)$ Give a geometric interpretation of the above integral by using a $3-$dimensional sketch. $(4)$

$($c$)$ Transform the above integral to a double integral with polar coordinates $($Do not evaluate the integral$).$

$2$