Final answer:
To find the volume of the solid bounded by the paraboloid and the plane in polar coordinates, we need to express the equations in terms of polar coordinates. The volume can be calculated using a triple integral in polar coordinates with the appropriate bounds. Evaluating this integral will give us the volume of the solid.
Step-by-step explanation:
To find the volume of the solid bounded by the paraboloid and the plane in polar coordinates, we need to express the equations in terms of polar coordinates. The paraboloid equation can be expressed as z = 4 + 2r^2, where r is the distance from the origin. The plane equation is z = 12. To find the bounds for the integral in polar coordinates, we set z = 4 + 2r^2 equal to z = 12 and solve for r. This gives us r = √4 = 2. Now we can set up the triple integral in polar coordinates to find the volume: V = ∫∫∫ r dz dr dθ, where the bounds for r are from 0 to 2, the bounds for θ are from 0 to π/2, and the bounds for z are from 4 + 2r^2 to 12. Evaluating this integral will give us the volume of the solid.