If you cut a hemisphere in half the two parts are not hemispheres so this method doesn't work.When r=a, the d=a, as the bowl is hemisphere, d=r.
Therefore, when d=0.5a, r=0.5a.
Volume of a hemisphere
=\frac{2}{3}\pi r^3
=\frac{\pi a^3}{12}
I would use the equation for a circle and rotate the region between a/2 and a around the x-axis.