Final answer:
To find the volume of the given solid, set up a triple integral in the cylindrical coordinate system using the equation of the paraboloid in polar coordinates. Evaluate the integral over the region in the xy-plane where the paraboloid intersects the plane z = 0.
Step-by-step explanation:
To find the volume of the given solid using polar coordinates, we can set up a triple integral in the cylindrical coordinate system. The equation of the paraboloid is given as z = 32 - 2x^2 - 2y^2. To convert this equation into cylindrical coordinates, we use the following substitutions: x = rcos(theta) and y = rsin(theta). Substituting these values into the equation of the paraboloid, we get z = 32 - 2r^2(cos^2(theta) + sin^2(theta)).
Since we want to find the volume above the xy-plane, the lower limit of z would be 0. Therefore, the integral to find the volume can be set up as follows:
V = ∫∫∫ r dz dr d(theta) over the region in the xy-plane where the paraboloid intersects the plane z = 0.
Simplifying this integral and evaluating it will give us the volume of the given solid.