Question

Find the volume of the solid enclosed by the paraboloid

x²+y²=2z, the xy-plane, and the cylinder x²+y²=4.

Answer 1

The volume of the solid enclosed by the paraboloid x²+y²=2z, the xy-plane, and the cylinder x²+y²=4 is 12π cubic units.

Step-by-step explanation:

To find the volume of the solid enclosed by the paraboloid x²+y²=2z, the xy-plane, and the cylinder x²+y²=4, we can use the method of cylindrical shells.

We will integrate over the region in the xy-plane defined by the cylinder x²+y²=4.

The volume is given by the integral of the height function (which is the difference between the paraboloid and the xy-plane) multiplied by the circumference of the cylinder.

After evaluating the integral, the volume of the solid is equal to 12π cubic units.