Problem 34 - Thomas Calculus

Question

Write inequalities to describe the sets in Exercises \(29-34\) The closed region bounded by the spheres of radius 1 and radius 2 centered at the origin. (Closed means the spheres are to be included. Had e wanted the spheres left out, we would have asked for the open region bounded by the spheres. This is analogous to the way we use closed and open to describe intervals: closed means endpoints included, open means endpoints left out. Closed sets include boundaries; open sets leave them out.)

Step-by-Step Solution

Answer

The region is described by \\(1 \leq x^2 + y^2 + z^2 \leq 4\\).

Step 1: Understanding the Region

We are asked to describe a closed region bounded by two spheres. These spheres are centered at the origin, with radii 1 and 2, respectively. A closed region includes the boundary, meaning the surfaces of the spheres are also part of the region.

Step 2: Sphere Equations

The equation for a sphere centered at the origin with radius \(r\) is given by \(x^2 + y^2 + z^2 = r^2\). For the spheres in the problem, we have: \(x^2 + y^2 + z^2 = 1\) for the smaller sphere and \(x^2 + y^2 + z^2 = 4\) for the larger sphere.

Step 3: Translating to Inequalities

To include the volume of the region inside and on the smaller sphere, we use \(x^2 + y^2 + z^2 \geq 1\). For the region inside and on the larger sphere, we use \(x^2 + y^2 + z^2 \leq 4\).

Step 4: Forming the Complete Set Description

Now, we combine these inequalities to describe the set that includes all points between and on the given spheres. Thus, the inequalities that describe the region are: \(1 \leq x^2 + y^2 + z^2 \leq 4\).

Key Concepts

Spheres

In three-dimensional space, a sphere is a perfectly symmetrical object where every point on its surface is equidistant from a central point, known as the center. The distance from the center to any point on the sphere is the radius. This concept is widely used in mathematics to describe objects with constant curvature.

For a sphere centered at the origin in a coordinate system, the equation can be written as:

• \[x^2 + y^2 + z^2 = r^2\]

Substituting any value for \(r\) gives us the corresponding radius of the sphere. Bigger \(r\) means a larger sphere. In our exercise, the spheres have radii of 1 and 2, centered at the origin, thus forming concentric spheres.

Closed Region

A closed region in geometry includes all the points inside a given shape, as well as the points on its boundary. This concept is similar to a closed interval in a one-dimensional line. In both cases, all endpoints or boundary points are included.

In the context of spheres, a closed region bounded by two concentric spheres includes every point from the surface of the smallest sphere to the surface of the largest sphere. This means:

• The closed region takes into account all interior points and includes both spherical surfaces.
• For our exercise, this involves the space where the inequality \(1 \leq x^2 + y^2 + z^2 \leq 4\) holds true.

Radius

The radius of a sphere is a crucial concept because it determines the size of the sphere. A radius is defined as the distance from the center of the sphere to any point on its surface. In the context of the exercise, the spheres in question have radii of 1 and 2.

This has several implications:

• The smaller sphere with radius 1 encapsulates a smaller volume about the origin compared to the larger sphere with radius 2.
• The equations \(x^2 + y^2 + z^2 = 1\) and \(x^2 + y^2 + z^2 = 4\) represent the surfaces of these spheres respectively.

Understanding these radii helps in visualizing the space that falls within and on the boundary of these spheres.

Boundary Inclusion

Boundary inclusion is an important aspect when describing closed sets. It refers to the inclusion of the boundary points in the region or set description.

When a region is specified as closed, it means all boundary surfaces are included in the region. In the problem outlined, both spheres' surfaces are included, signifying a complete closed region.

• This impacts how the inequalities are written:
  • For the region to include the surface of the smaller sphere, we use \(x^2 + y^2 + z^2 \geq 1\).
  • To include the surface of the larger sphere, we write \(x^2 + y^2 + z^2 \leq 4\).
  • Thus, the inequality that captures boundary inclusion is \(1 \leq x^2 + y^2 + z^2 \leq 4\).

Being thorough with boundary inclusion ensures a complete understanding of the region described by the inequalities.