Solid inside both x^2+y^2+z^2=a^2 and (x-(a/2)^2 +y^2 = (a/2)^2
You need to first visualize what the solid looks like. In this case, the first equation describes a sphere of radius a centered at the origin. For the second, I assume you mean (x-(a/2))^2 +y^2 = (a/2)^2, so it is a cylinder with radius a/2 centered at (a/2,0).
It will be easiest to integrate with respect to z first. The top and bottom surfaces come from the sphere, so solve that equation for z to get the limits of integration.
We now need to integrate this over the circle (x-(a/2))^2 +y^2 = (a/2)^2. So you solve for y to get the limits for y: .
The limits for x are just the right- and left-most points of the circle, 0 and a.