5
You want the volume above the paraboloid below the sphere, so first observe that z goes from the paraboloid to the sphere. Let's use cylindrical coordinates:
x=rcosθ
y=rsinθ
z=z
Then dV=rdzdrdθ and we have so far
∫∫∫2−r2√r2rdzdrdθ.
To find the limits on r and θ project the curve of intersection of the given surfaces, onto the x−y plane. It is immediate that the region is a circle of radius 1 so we see that r goes from 0 to 1 and θ goes from 0 to 2π. Thus, we have
∫2π0∫10∫2−r2√r2rdzdrdθ=π6(82–√−7)