Question

Describe in words and then sketch the solid region that satisfies both 1 ≤ x² + y² + z² ≤ 2 and y ≤ 0.

Answer 1

The solution represents a solid region within two concentric spheres of radii 1 and √2 and includes only the half-space below the xy-plane where y is negative.

Step-by-step explanation:

The given conditions describe a solid region in three-dimensional space that is between two spheres and below the xy-plane. The inequalities 1 ≤ x^2 + y^2 + z^2 ≤ 2 define a region that lies between the sphere with radius 1 and the sphere with radius √2. Since we also have y ≤ 0, this region is only the half that is below the xy-plane (where y values are negative).

To sketch this, we start by drawing the larger sphere with radius √2, then the smaller sphere with radius 1, and finally, we shade the region between them that is below the xy-plane where y is negative.