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 equa
tion 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
