Problem 21 - Thomas Calculus

Question

In Exercises \(17-24,\) describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. $$ \begin{array}{l}{\text { a. } 1 \leq x^{2}+y^{2}+z^{2} \leq 4} \\ {\text { b. } x^{2}+y^{2}+z^{2} \leq 1, \quad z \geq 0}\end{array} $$

Step-by-Step Solution

Answer

a: Spherical shell between radii 1 and 2; b: Upper hemisphere of radius 1.

Step 1: Analyzing the inequality in part a

For part (a), the inequality \(1 \leq x^2 + y^2 + z^2 \leq 4\) describes the set of points in space that lie between two spheres. The first sphere has a radius of \(1\) and is centered at the origin. The second sphere has a radius of \(2\) and is also centered at the origin. The inequality means that the points are on or outside the first sphere and on or inside the second sphere. This describes a spherical shell or annular region in space.

Step 2: Analyzing the inequalities in part b

For part (b), the inequality \(x^2 + y^2 + z^2 \leq 1\) describes a solid sphere with radius \(1\) centered at the origin. The additional condition \(z \geq 0\) restricts this solid sphere to the upper hemisphere, meaning points with non-negative \(z\)-values only. Thus, the solution is the upper half of a sphere with radius \(1\).

Key Concepts

Understanding the Spherical Shell

In three-dimensional space, when we talk about a spherical shell, we are referring to a region between two concentric spheres. In exercise part (a), you encountered the inequality \(1 \leq x^2 + y^2 + z^2 \leq 4\). This means that all the points satisfying this inequality are located between two concentric spheres.
This forms a spherical shell or a hollow sphere-like layer.

• The inner sphere has a radius of \(1\), meaning any point at a distance greater than \(1\) from the origin is included in the shell.
• The outer sphere has a radius of \(2\), including everything within or on its surface.
• Thus, the shell includes points that are not inside the inner sphere but are contained within the outer sphere.

This concept is essential in visualizing three-dimensional regions bounded by specific geometric parameters.

Exploring the Solid Sphere

A solid sphere represents a fully filled spherical volume in 3D space. It includes every point from the center of the sphere out to its surface. For the inequality \(x^2 + y^2 + z^2 \leq 1\) in part (b) of the exercise, this describes a solid sphere with a radius of \(1\).

• Every point within this sphere satisfies the condition that the distance from the origin is less than or equal to \(1\).
• The sphere is centered at the origin \(0,0,0\).
• This sphere is a fundamental example of spatial constraints and helps in understanding volumes contained by specific inequalities.

The solid sphere is a core shape in geometry and physics, often used to model closed systems or objects.

Focusing on the Upper Hemisphere

The term upper hemisphere refers to the half of a sphere that lies above a specified plane, typically the plane \(z = 0\). In the context of the exercise, part (b) further refines the solid sphere by adding the condition \(z \geq 0\), which confines the points to those with non-negative \(z\) coordinates.

• This means you only consider the top half of the sphere, forming a dome-like shape above the \(xy\)-plane.
• All points on the plane itself can be included, assuming they meet the previous inequality \(x^2 + y^2 + z^2 \leq 1\).
• The upper hemisphere is often used when focusing on phenomena occurring in positive vertical directions or contexts.

Understanding hemispheres is crucial in problems involving symmetry and divisibility of spherical shapes.