The of the solid region that is inside the cylinder , below the paraboloid and above the plane (assuming ) is which of the following?
D) 6
E) None of the above

Question

The  of the solid region that is inside the cylinder  , below the paraboloid  and above the  plane (assuming  ) is which of the following?
D) 6 
E) None of the above

Quizplus · Accepted Answer

The given solid region is bounded below by the xy-plane and above by the paraboloid z = x^2 + y^2 and inside the cylinder x^2 + y^2 = 4. 
Converting to cylindrical coordinates, we get,
z = r^2, and the cylinder is r = 2
The limits of integration for r are from 0 to 2.
The limits of integration for θ are from 0 to 2π. 
The limits of integration for z are from 0 to r^2.
Hence, the triple integral to find the volume is: 
∫[0,2] ∫[0,2π] ∫[0,r^2] r dz dθ dr
= ∫[0,2] ∫[0,2π] r^3/3 dθ dr
= 4π/3, which is option B.

The of the Solid Region That Is Inside the Cylinder