Problem 25 - Essential Calculus Early Transcendentals

Question

Describe in words the region of \(\mathbb{R}^{3}\) represented by theequations or inequalities. 0\(\leqslant z \leqslant 6\)

Step-by-Step Solution

Answer

The region is a slab in \(\mathbb{R}^{3}\) between the planes z=0 and z=6, extending infinitely in the x and y directions.

Step 1: Understanding the Coordinate System

In three-dimensional space, represented as \(\mathbb{R}^3\), we use three axes: the x-axis, the y-axis, and the z-axis. Points in this space are given as \((x, y, z)\). The z-axis typically represents the vertical position.

Step 2: Analyzing the Inequality

The inequality \(0 \leq z \leq 6\) specifies the bounds for z. This means the variable z can take any value between 0 and 6, inclusive. There are no restrictions on x or y, so they can take any real number values.

Step 3: Describing the Region

Given the inequality, the region described is where the z-coordinate, for any point \((x, y, z)\), must be between 0 and 6. Graphically, this region is a slab or a layer that extends infinitely in the x and y direction, but is confined vertically between z = 0 and z = 6.

Step 4: Conclusion

Thus, the region represents all the space between two parallel planes: one at z = 0 and the other at z = 6. This includes the planes themselves. There are no restrictions on the x or y coordinates, so they fill the entire space horizontally.

Key Concepts

Coordinate System in Three-Dimensional Space

In three-dimensional space, often denoted as \( \mathbb{R}^3 \), we describe positions using a coordinate system comprised of three axes: the x-axis, the y-axis, and the z-axis. Each point in this space is described by three numbers, \((x, y, z)\), where:

• The x-coordinate specifies the position along the horizontal x-axis.
• The y-coordinate specifies the position along another horizontal axis perpendicular to the x-axis, called the y-axis.
• The z-coordinate specifies the position along a vertical axis known as the z-axis.

This coordinate system allows us to locate any point in three-dimensional space by assigning a unique set of three values. For instance, the point \((3, 2, 5)\) is 3 units along the x-axis, 2 units along the y-axis, and 5 units up the z-axis. This system helps visualize and analyze the spatial relationships between different objects and regions.

Understanding Inequalities in \( \mathbb{R}^3 \)

Inequalities involving coordinates help define what values a particular coordinate can take within a three-dimensional space. Let's take a closer look at the inequality \(0 \leq z \leq 6\).

• This specific inequality sets the bounds for the z-coordinate. It means that z can be any number from 0 to 6, inclusive of both 0 and 6.
• Notice that there are no restrictions on the x and y-coordinates here, meaning they can be any real number. So, x and y can take any value from negative to positive infinity.

By using such inequalities, we can define specific segments of three-dimensional space that meet certain conditions. This also means we can gauge where objects or points are allowed to exist within those bounds. Such understanding is crucial in fields that demand spatial cognition like engineering, physics, and computer graphics.

Graphical Representation of a Bounded Region

Visualizing the effects of inequalities in three-dimensional space can help in comprehensively understanding the defined region. In the example \(0 \leq z \leq 6\), we are describing a region along the z-axis:

• Graphically, this inequality describes a slab or layer within \( \mathbb{R}^3 \) that stretches infinitely in the x and y directions but is vertically confined between the planes at z = 0 and z = 6.
• The use of parallel planes at these z-values vividly illustrates the bounds—they form the top and bottom surfaces of our slab.
• This slab includes every point where the z-coordinate meets the inequality condition, meaning anywhere the points are exactly on, above, or below between these planes.

Such graphical representations are invaluable tools for better understanding the spatial extent of regions dictated by inequalities. They allow us to easily conceptualize volume and space in both theoretical and practical applications.