Question

Find the volume of the solid bounded by the plane z = 0 and the paraboloid z = 1 - x2 - y2. SOLUTION If we put z = 0 in the equation of the paraboloid, we get x2 + y2 = 1, so the solid lies under the paraboloid and above the circular disk D given by x2 + y2 ≤ 1. In polar coordinates D is given by 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π. Since 1 - x2 - y2 = 1 - r2, the volume is

Answer 1

The student is determining the volume of a solid using polar coordinates where the solid is bounded by a plane and a paraboloid.

Step-by-step explanation:

The student is tasked with finding the volume of a solid bounded by the plane z = 0 and the paraboloid defined by z = 1 - x2 - y2. To approach this problem, the use of polar coordinates is recommended. By setting z = 0, we determine that the solid lies above the disk D, where x2 + y2 ≤ 1. In polar coordinates, D is expressed as 0 ≤ r ≤ 1 and 0 ≤ θ ≤ 2π. The volume can be calculated by integrating the function 1 - r2 over the disk D using a double integral in polar coordinates.

Answer 2

3

Step-by-step explanation: