$($Section $14\mathrm{.}6,$ Chapter $17)$

Let $V$ be a region in $R^3$ bounded above by the hemisphere $z=\sqrt[\phantom{2}]{1-x^2-y^2}$ and below by the cone $z=\sqrt[\phantom{2}]{x^2+y^2}-1.$ Let $S$ be the surface of $V($consisting of the hemisphere on top and the paraboloid below$).$

$($a$)$ Compute the volume of $V$ using spherical coordinates.

$(10)$

$($b$)$ Sketch $S$ and the $xY-$projection of $S.$

$(3)$

$($c$)$ Use a surface integral to evaluate the area of $S.$

$(8)$

$[21]$