/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Sketch the region described by t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 7

Sketch the region described by the following cylindrical coordinates in three- dimensional space. $$r^{2}+z^{2}=4$$

Short Answer

Expert verified
The region is a cylinder of radius 2 around the \( z \)-axis.

Step by step solution

01

Understanding Cylindrical Coordinates

Cylindrical coordinates are a way of describing the location of a point in three-dimensional space using a radius \( r \), an angle \( \theta \), and a height \( z \). In this problem, the variable \( r \) represents the distance from the point to the \( z \)-axis, and \( z \) is the vertical distance from the \( xy \)-plane.
02

Rewrite the Equation

The given equation is \( r^2 + z^2 = 4 \). This is similar in form to the equation of a circle in the vertical \( rz \)-plane. Rewrite it in standard form to better understand it: \( r^2 + z^2 = 2^2 \). This indicates that the radius of the circle in the \( rz \)-plane is 2.
03

Visualize the Shape of the Region

In cylindrical coordinates, this equation describes a circular cross-section of radius 2 in any slice parallel to the \( rz \)-plane. Imagine rotating this circle around the \( z \)-axis. This means the region forms a three-dimensional shape.
04

Identify the Solid of Revolution

The described region is a right circular cylinder with radius of 2. The circle \( r^2 + z^2 = 4 \) rotated around the \( z \)-axis forms the lateral surface of the cylinder.
05

Conclusion and Sketch

Sketch a vertical line representing the \( z \)-axis. Around this axis, draw a cylinder with a radius of 2 extending infinitely in both directions along the \( z \)-axis. This is the described region for the given equation.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Three-Dimensional Space
In mathematics, three-dimensional space is a geometric representation of the world around us, characterized by three axes: the x-axis, y-axis, and z-axis. Each point in this space can be pinpointed using three coordinates. Here, the cylindrical coordinate system is particularly useful when dealing with objects or regions that possess symmetry around a central axis.
  • The x and y coordinates locate the position on a flat plane, much like positioning a point on a map.
  • The z coordinate, however, adds depth, allowing you to determine how high or low a point is from that plane.
Three-dimensional space enables us to model and visualize objects with volume and depth, essential for engineering, physics, and many branches of mathematics.
Circle in the rz-plane
The concept of an rz-plane arises when working with cylindrical coordinates, where one examines a specific slice of the three-dimensional space. Here, the rz-plane helps in visualizing cross-sections of volumes.
  • In this plane, r denotes the radial distance from the z-axis, much like the radius of a circle perpendicular to the z-axis.
  • In our exercise, the equation \(r^2 + z^2 = 4\) represents a circle of radius 2.
By interpreting this equation correctly, one can identify a circle in the rz-plane, which is pivotal for understanding the larger implications when this circle is revolved around an axis to form a solid.
Solid of Revolution
A solid of revolution is created when a two-dimensional shape is rotated around an axis. In our context, consider the circle given by \(r^2 + z^2 = 4\) in the rz-plane. When this circle is revolved around the z-axis, it generates what mathematicians call a solid of revolution.
  • This process involves imagining the two-dimensional circle spinning along the axis, creating a surface or volume through its path.
  • The resulting three-dimensional object can have varying forms, including spheres, cylinders, and toroids, depending on the initial shape and the axis of rotation.
Understanding solids of revolution is crucial for problem-solving in calculus and physics, where volumes of such shapes often need to be calculated.
Right Circular Cylinder
The right circular cylinder is a common shape encountered in geometry, characterized by straight parallel sides and circular bases. In the context of our exercise, the region described by the equation \(r^2 + z^2 = 4\) results in a right circular cylinder upon full revolution around the z-axis.
  • The radius of this cylinder is 2, matching the radius of the circle in the rz-plane.
  • It extends infinitely along the z-axis, since there are no bounds given for z, making it an infinitely tall cylinder.
Understanding the formation and properties of right circular cylinders helps in applications ranging from architectural design to the modeling of cylindrical containers in practical engineering scenarios.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The integrals we have seen so far suggest that there are preferred orders of integration for cylindrical coordinates, but other orders usually work well and are occasionally easier to evaluate. Evaluate the integrals. $$\int_{-1}^{1} \int_{0}^{2 \pi} \int_{0}^{1+\cos \theta} 4 r\quad d r\quad d \theta \quad d z$$

Find the volume of the solid cut from the square column \(|x|+|y| \leq 1\) by the planes \(z=0\) and \(3 x+z=3\)

A solid is bounded below by the cone \(z=\sqrt{x^{2}+y^{2}}\) and above by the plane \(z=1 .\) Find the center of mass and the moment of inertia about the \(z\) -axis if the density is a. \(\delta(r, \theta, z)=z\) b. \(\delta(r, \theta, z)=z^{2}\)

Let \(D\) be the region bounded below by the cone \(z=\sqrt{x^{2}+y^{2}}\) and above by the paraboloid \(z=2-x^{2}-y^{2} .\) Set up the triple integrals in cylindrical coordinates that give the volume of \(D\) using the following orders of integration. a. \(d z d r d \theta\) b. \(d r d z d \theta\) c. \(d \theta d z d r\)

Evaluate the spherical coordinate integrals. $$\int_{0}^{2 \pi} \int_{0}^{\pi / 3} \int_{\sec \phi}^{2} 3 \rho^{2} \sin \phi d \rho d \phi d \theta$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.