Problem 16 - Thomas Calculus

Question

In Exercises \(13-18\) , describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. $$ \begin{array}{l}{\text { a. } x^{2}+y^{2} \leq 1, \quad z=0 \quad \text { b. } x^{2}+y^{2} \leq 1, \quad z=3} \\ {\text { c. } x^{2}+y^{2} \leq 1, \text { no restriction on } z}\end{array} $$

Step-by-Step Solution

Answer

a. Filled circle (disc) on xy-plane; b. Filled circle (disc) at z=3; c. Infinite cylinder centered at origin with radius 1.

Step 1: Understand the Equalities and Inequalities

For each condition, recognize the geometric shapes represented by the equations and inequalities given. The inequality \( x^2 + y^2 \leq 1\) describes a circle of radius 1 centered at the origin in the xy-plane. The equation \( z = 0 \) fixes the position on the z-coordinate, similarly for \( z = 3 \). The condition that there are no restrictions on \( z \) indicates freedom in the z-axis direction.

Step 2: Describe the Region for Part (a)

For part (a), we have \( x^2 + y^2 \leq 1 \) and \( z = 0 \). This represents a filled circle (a disc) of radius 1 centered at the origin, lying on the xy-plane.

Step 3: Describe the Region for Part (b)

For part (b), we have \( x^2 + y^2 \leq 1 \) and \( z = 3 \). This again represents a filled circle (a disc) of radius 1, but this time it is parallel to the xy-plane at the height \( z = 3 \).

Step 4: Describe the Region for Part (c)

For part (c), we have \( x^2 + y^2 \leq 1 \) with no restriction on \( z \). This describes an infinite cylindrical region where each cross-section parallel to the xy-plane is a circle of radius 1.

Key Concepts

Inequalities in 3D

When dealing with inequalities in 3D, you're exploring regions in a three-dimensional space. Instead of finding a single point, you're often identifying areas or volumes that satisfy certain conditions. For instance, consider the inequality \(x^2 + y^2 \leq 1\). This represents all the points where the distance from the origin on the xy-plane is less than or equal to 1. In 3D, such an inequality represents a filled circular region, also known as a disc.

• This disc is centered at the origin and lies on the xy-plane.
• The boundary of this disc is defined by the circle \(x^2 + y^2 = 1\), and includes every point inside of it.

Understanding these inequalities enables you to visualize complex shapes in three dimensions, marking all possible points that fulfil the conditions.

Cylinders in 3D

Cylinders can take many forms in 3D geometry. A basic cylindrical shape is defined by its circular cross-section and extends infinitely in one direction. When you have an equation like \(x^2 + y^2 \leq 1\), without any restriction on \(z\), you're describing a cylinder that stretches infinitely upwards and downwards along the z-axis. This particular cylinder has its base centered on the xy-plane.

• Every cross-section parallel to the xy-plane is a circle of radius 1.
• It extends infinitely along the z-axis.

This concept of cylinders helps in understanding various real-world structures and is essential in engineering and architectural design.

Coordinate Systems in 3D

Understanding coordinate systems in 3D is fundamental when working with equations and inequalities in three-dimensional space. The common coordinate system used is the Cartesian coordinate system, which uses three perpendicular axes: the x-axis, y-axis, and z-axis. Each point in space can be identified using a tuple \((x, y, z)\).

• The x and y coordinates represent a point on a plane, while the z coordinate indicates height or depth.
• A point \((x, y, z)\) in 3D defines a unique position relative to the origin where the three axes meet.

Understanding this system allows you to visualize 3D spaces effectively and provides a basis for plotting shapes, solving spatial equations, and modeling real-world structures. With this knowledge, you can interpret and visualize various geometric configurations in three dimensions.