Agree that is the region consisting of everything on and inside the sphere
       
as plotted above, and calculate the integral
       .

Answer:

Look at the plotting code:
       
                     ,
                      .
And go with
       ;
with
       , ,
and .
This gives you
       
       
       
       .

You gotta integrate with respect to before you integrate with respect to because the limit of integration with respect to changes as changes.

Now calculate the volume conversion factor:

Calculate
       
       .

Done.