/*! 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 11 Identify and sketch the followin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 11

Identify and sketch the following sets in cylindrical coordinates. $$\\{(r, \theta, z): 0 \leq r \leq 3,0 \leq \theta \leq \pi / 3,1 \leq z \leq 4\\}$$

Short Answer

Expert verified
The given cylindrical coordinates describe a cylindrical-cut sector with a base angle of 60 degrees and a vertical thickness of 3 units, between heights z = 1 and z = 4.

Step by step solution

01

Identify the shape's properties in terms of r, 胃, and z.

Given cylindrical coordinates are: $$ 0 \leq r \leq 3,0 \leq \theta \leq \pi/3,1 \leq z \leq 4 $$ They describe: - The range of the radius (r) is from 0 to 3. - The range of the angle (胃) is from 0 to \(\pi/3\). - The range of the height (z) is from 1 to 4.
02

Identify the shape.

Combining the ranges for r, 胃, and z, we can conclude: - The shape will be centered at the origin, since r starts at 0. - The shape extends radially outwards by a distance of 3 units, so we can imagine a cylindrical "cut" with a radius of 3 units. - The shape is limited between 0 and \(\pi/3\) in the xy-plane, which corresponds to a \(60^\circ\) circular sector in the xy-plane. - The shape is limited in the vertical z-direction between heights 1 and 4, so it has a thickness of 3 units in the vertical direction. Considering these properties, the given set describes a "cut" of a cylinder, where the circular base has a 60-degree sector, and the height spans from 1 to 4.
03

Sketch the shape.

To sketch the given set: 1. Draw the right rectangular coordinate system (x, y, z). 2. Mark the range of the angle (胃) in the xy-plane, which is 0 to \(\pi/3\), or 0 to \(60^\circ\). 3. Draw the arc of radius 3 at \(60^\circ\) in the xy-plane. 4. At z = 1 and z = 4, draw two horizontal lines parallel to the xy-plane. 5. Connect the points on the arc at z = 1 to the corresponding points on the arc at z = 4, forming the curved surface of the cylindrical cut. 6. Fill in the sketch to represent the solid shape. After sketching, you should have a cylindrically-cut sector with a base angle of 60 degrees and a vertical thickness of 3 units, between heights z = 1 and z = 4.

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!

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

Evaluate the following integrals in spherical coordinates. $$\int_{0}^{2 \pi} \int_{0}^{\pi / 3} \int_{0}^{4 \sec \varphi} \rho^{2} \sin \varphi d \rho d \varphi d \theta$$

Use spherical coordinates to find the volume of the following solids. That part of the ball \(\rho \leq 4\) that lies between the planes \(z=2\) and \(z=2 \sqrt{3}\)

Use a change of variables to evaluate the following integrals. $$\begin{aligned} &\iiint_{D} x y d V ; D \text { is bounded by the planes } y-x=0\\\ &y-x=2, z-y=0, z-y=1, z=0, \text { and } z=3 \end{aligned}$$

Suppose a wedge of cheese fills the region in the first octant bounded by the planes \(y=z, y=4,\) and \(x=4\) You could divide the wedge into two pieces of equal volume by slicing the wedge with the plane \(x=2 .\) Instead find \(a\) with \(0

Let \(R\) be the region bounded by the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1,\) where \(a>0\) and \(b>0\) are real numbers. Let \(T\) be the transformation \(x=a u, y=b v\) Find the average square of the distance between points of \(R\) and the origin.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.