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