Answer
$\dfrac{2}{3} \pi R^3$
Work Step by Step
Re-write as: $z=\sqrt {R^2-x^2-y^2} $
or, $ z^2=R^2-x^2-y^2$
or, $ x^2+y^2+z^2 =R^2$
The equation of an upper half of a hemi-sphere can be written as: $x^2+y^2+z^2 =R^2$ when $z \geq 0$
So, we can find the volume as follows:
$Volume =(\dfrac{1}{2}) (\dfrac{4}{3} \pi R^3) =\dfrac{2 \pi R^3}{3}$