Final answer:
To find the volume of the solid enclosed by the paraboloid z=6x² +6y² and the plane z=6, we can set up a triple integral in Cartesian coordinates. The region of integration for x and y is the unit circle in the xy-plane. The triple integral to find the volume is ∭(0 to 6)∭(0 to 2π)∭(0 to 1) dz dθ dr.
Step-by-step explanation:
We can find the volume of the solid enclosed by the paraboloid and the plane by setting up a triple integral in Cartesian coordinates. The paraboloid is given by the equation z = 6x² + 6y², and the plane is given by z = 6. To find the volume, we need to integrate over the region where the paraboloid is above the plane, which corresponds to the values of z between 0 and 6. The limits for x and y can be determined by the intersection of the paraboloid and the plane.
Let's solve for the intersection:
6x² + 6y² = 6
x² + y² = 1
This equation represents a circle with radius 1 in the xy-plane. Therefore, the region of integration for x and y is the unit circle in the xy-plane.
The triple integral to find the volume is:
∭(0 to 6)∭(0 to 2π)∭(0 to 1) dz dθ dr