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Chapter 12: Problem 45

In Problems 45 through 52 . describe and sketch the surface or solid described by the given equations and/or inequalities. \(r=3,-1 \leqq z \leqq 1\)

Short Answer

Expert verified
This describes a finite cylinder with radius 3 and height extending from \( z = -1 \) to \( z = 1 \).

Step by step solution

01

Identify the Surface

The given equation is in cylindrical coordinates, where \( r = 3 \) describes a cylindrical surface centered on the z-axis with a radius of 3. The equation is not dependent on \( \theta \), indicating it extends around the entire circle of radius 3.
02

Apply the Inequality

The inequality \(-1 \leq z \leq 1\) restricts the height of the cylinder along the \( z \)-axis. This means the cylinder extends from \( z = -1 \) to \( z = 1 \).
03

Sketch the Surface

To sketch the described solid, draw a cylinder centered at the origin (0,0) in the \( xy \)-plane with a radius of 3. The vertical extent of the cylinder runs between the \( z \)-planes at \( z = -1 \) and \( z = 1 \). This creates a finite-height cylindrical surface.
04

Describe the Solid

The surface is a finite cylinder with base circles in the \( xy \)-plane having radius 3 and the top and bottom base at \( z = 1 \) and \( z = -1 \), respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cylinder
A cylinder is a three-dimensional surface formed by translating a closed curve, most commonly a circle, along an axis. In this context, the axis is the z-axis. The cylinder described here is formed by revolving a circle of radius 3 around the z-axis, extending indefinitely if not for the limits set by the height.
  • Has a circular base, typically defined in the xy-plane.
  • Extends along the z-axis; can be infinite or restricted to a specific range.
  • In cylindrical coordinates, it's often described by fixing the radius, here given by \( r = 3 \).
Cylinders are essential structures in various fields, enabling us to represent pipes, columns, or even the basic structure of a can. By visualizing this shape, one can anticipate how it might appear or be applied.
Radius
The radius in any context is the distance from the center of a circle to its outer edge. In this problem, the radius is crucial as it clearly defines how wide the cylindrical surface is from the axis.
  • Here, the radius is fixed at 3, meaning every point on the surface is exactly 3 units away from the z-axis.
  • This radius dictates the circular size of the base in the xy-plane.
  • A consistent radius maintains the cylinder's uniform width.
Understanding the concept of radius helps in visualizing the cylinder's bases. It plays a significant role in calculations involving area or volume of cylindrical surfaces, often appearing squared in such formulas due to involvement in base area calculations like \( \pi r^2 \).
Inequality
Inequalities in this context help us define and limit the regions in space where the cylinder is present. The inequality \(-1 \leq z \leq 1\) defines the height of the cylinder.
  • Dictates the boundary of the cylinder along the z-axis.
  • Z extends from -1 to 1, creating a finite height rather than an infinite extension.
  • Allows us to think of the cylinder as having a top and bottom at these z-values.
This restriction is crucial for understanding that although the cylinder has indefinite circular reach horizontally, it has defined vertical limits. Understanding inequalities in 3D space helps in geometry and engineering, ensuring structures fit within designated spatial constraints.
Cylindrical Surface
A cylindrical surface is a three-dimensional structure made by revolving a line around an axis, not intersecting it. Here, we see a cylindrical surface with a determined radius and height.
  • The equation \( r = 3 \) characterizes the cylindrical surface around a central axis.
  • Not dependent on \( \theta \), it suggests a seamless, rounded surface.
  • The surface is capped between \( z = -1 \) and \( z = 1 \).
Visualizing the cylindrical surface helps in associating theoretical equations with physical shapes. Recognizing such surfaces enables better comprehension in practical applications such as physics and architectural design.

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Most popular questions from this chapter

Convert the given equation both to cylindrical and to spherical coordinates. \(x+y+z=1\)

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