Answer
Hemisphere in the first quadrant.
Work Step by Step
Since
$$ x=r\cos \theta,\quad y=r\cos \theta, \quad z=z,$$
then $$ x^2+y^2+z^2=4$$
takes the form
$$ r^2\cos^2\theta+r^2\sin^2\theta+z^2=r^2+z^2= 4, \quad 0\leq\theta\leq \pi/2.$$
Which is a hemisphere in the first quadrant.
