Problem 31 - Essential Calculus Early Transcendentals

Question

The region between the \(y z\) -plane and the vertical plane \(x=5\)

Step-by-Step Solution

Answer

The region is \((x, y, z)\) where \(0 \leq x \leq 5\), for any \(y, z\).

Step 1: Understanding the Problem

We are identifying a region in 3D space that is bounded by two different planes: the \( yz \)-plane and the vertical plane where \( x = 5 \). The \( yz \)-plane corresponds to the plane where \( x = 0 \), whereas the second plane is parallel to the \( yz \)-plane but located at \( x = 5 \). The objective is to describe this region in mathematical terms.

Step 2: Define the Region Boundaries

This region is bounded by two planes: the \( yz \)-plane defined by \( x = 0 \) and the plane \( x = 5 \). There are no constraints given on the \( y \) or \( z \) axes, implying that these values could span all real numbers. Therefore, the region is defined as all points \( (x, y, z) \) such that \( 0 \leq x \leq 5 \) for any real values of \( y \) and \( z \).

Step 3: Visualize the Region

You can visualize this region by imagining a "slab" of space that starts at the \( yz \)-plane and extends outwards until it reaches \( x = 5 \). This slab is infinite along the \( y \) and \( z \) directions, as there are no restrictions provided for these dimensions.

Key Concepts

yz-plane

The concept of the "\( yz \)-plane" is fundamental in the study of 3D spaces. Imagine a sheet of paper that stretches infinitely in the "upward" and "rightward" directions, that would be akin to the \( yz \)-plane in three-dimensional space. Here, every point is defined by coordinates \( (x, y, z) \) where the \( x \) coordinate is set to zero. This means any point on the \( yz \)-plane is represented by \((0, y, z)\).

• Z-axis: Represents the vertical direction on this plane.
• Y-axis: Represents the horizontal direction.

The \( yz \)-plane is one of the three primary coordinate planes used to describe all possible points in 3D space. Understanding this plane helps in solving problems involving spatial regions like the one in our exercise. It is a flat surface with no depth (in the direction of the x-axis).

3D visualization

"3D visualization" is key to grasping regions in three-dimensional space effectively. Imagine you have drawn a block or a slab in space that describes a region deemed necessary to understand through a given exercise.

Visualizing the space from steps described in a problem, such as imagining a slab that lies between the \( yz \)–plane and another plane at \( x = 5 \), can greatly enhance comprehension. It helps bridge abstract mathematical descriptions and more tangible, visual concepts.

• Imagine two planes enveloping the space, starting from where \( x = 0 \) up to \( x = 5 \).
• Picturing it as an endless slab extending infinitely in both the y- and z-directions can clarify the extent of the region.

With visualizations, you can see the transitions and span of axes necessary to solve and visualize equations and inequalities involved in 3D space mathematics.

coordinate planes

"Coordinate planes" are planes that are defined by holding one of the three coordinate values constant in 3D space. There are three primary coordinate planes:

• xy-plane: Defined by \( z = 0 \), it extends across the x-axis and y-axis.
• yz-plane: Defined by \( x = 0 \), covers the y-axis and z-axis.
• zx-plane: Defined by \( y = 0 \), spans the z-axis and x-axis.

These planes are crucial because they simplify the observation and understanding of spatial relationships in any 3D space configuration.

By visualizing these planes,

• You can break down complex 3D structures into manageable 2D sections.
• Coordinate planes help delineate areas, regions, and boundaries.

This understanding is used in many applications, from plotting points and sketching functions to solving intricate geometrical challenges within cubic space.