Answer
$x^{2} + y^{2} + z^{2} \leq 4$, $z\geq 0$
Work Step by Step
A sphere with radius 2 centered at the origin has the equation:
$x^{2} + y^{2} + z^{2} = 4$
All the points on or inside this sphere, which put together make a solid sphere, would then be represented by:
$x^{2} + y^{2} + z^{2} \leq 4$
If we only want the upper hemisphere of the sphere, we have to restrict the values of z to only nonnegative numbers. To do this we simply add the inequality
$z\geq 0$.
Therefore our solid upper hemisphere is described by:
$x^{2} + y^{2} + z^{2} \leq 4$
$z\geq 0$
