The same result as Luca M gave can also be accomplished leaving more thinking to Mathematica, using Boole, which allows defining region of integration through inequalities:

Integrate[Boole[x^2 + y^2 + z^2 < a^2 && y > a/2],

{y, -Infinity, Infinity}, {x, -Infinity, Infinity}, {z, -Infinity, Infinity},

Assumptions -> a > 0]

Mathematica is not always that clever integrating over implicit regions. Even in this case, it's best to use order of y, x, z instead of x, y, z to get result quickly.