/*! 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} Q 73. Let R聽be the radius of the base... [FREE SOLUTION] | 91影视

91影视

Find study content
Learning Materials

Discover learning materials by subject, university or textbook.

Explanations

Textbooks
All Subjects

Anthropology

Archaeology

Architecture

Art and Design

Bengali

Biology

Business Studies

Greek

Chemistry

Chinese

Combined Science

Computer Science

Economics

Engineering

English

English Literature

Environmental Science

French

Geography

German

History

Hospitality and Tourism

Human Geography

Italian

Japanese

Law

Macroeconomics

Marketing

Math

Media Studies

Medicine

Microeconomics

Music

Nursing

Nutrition and Food Science

Physics

Polish

Politics

Psychology

Religious Studies

Sociology

Spanish

Sports Sciences

Translation

All Subjects

Biology

Business Studies

Chemistry

Combined Science

Computer Science

Economics

English

Environmental Science

Geography

History

Math

Physics

Psychology

Sociology
Features
Features

Discover all of these amazing features with a free account.

Flashcards

91影视 AI

Notes

Study Plans

Study Sets
What鈥檚 new?

Flashcards
Study your flashcards with three learning modes.

Study Sets
All of your learning materials stored in one place.

Notes
Create and edit notes or documents.

Study Plans
Organise your studies and prepare for exams.
91影视
Discover

All the hacks around your studies and career - in one place.

Find a degree

Magazine
Featured

Magazine
Trusted advice for anyone who wants to ace their studies & career.

The largest student job board with the most exciting opportunities.

Verified student deals from top brands.

Discover our mobile app to take your studies anywhere.

Learning Materials

Features

Discover

Chapter 13: Q 73. (page 1068)

Let $R$ be the radius of the base of a cone and $h$ be the height of the cone. Use cylindrical coordinates to set up and evaluate a triple integral proving that the volume of the cone is $\frac{1}{3} 蟺 R^{2}$ .

Short Answer

Expert verified

This can be proved by taking an equation of cone that gives it a point at the origin that opens upward and downward.

Step by step solution

01

Given Information

It is given that $R$ is the radius of cone and $h$ is radius of cone.

02

Evaluate the limits

Let us assume equation $a^{2} z^{2} = h^{2} x^{2} + h^{2} y^{2}$ and it gives it a point at the origin that opens upward and downward and $h = z$ . It gives $a$ as radius of circle.

$鈮� 胃鈮� 2 蟺, 0 鈮� r 鈮� R, \frac{h}{R} 鈮� z 鈮� h$ is the region bounded by curves.

03

Calculating Volume of cone

The required volume is given by

$V = {鈭�}_{0}^{2 蟺} {鈭�}_{0}^{R} {鈭�}_{\frac{h r}{R}}^{h} r d z d r d 胃$

$V = {鈭�}_{0}^{2 蟺} {鈭�}_{0}^{R} (z)_{\frac{h r}{R}}^{h} r d r d 胃$

role="math" localid="1652391977550" $V = {鈭�}_{0}^{2 蟺} {鈭�}_{0}^{R} (h - \frac{h r}{R}) r d r d 胃$

$V = {鈭�}_{0}^{2 蟺} (h {鈭�}_{0}^{R} r d r - \frac{h}{R} {鈭�}_{0}^{R} r^{2} d r) d 胃$

Simplifying further

$V = {鈭�}_{0}^{2 蟺} (\frac{h R^{2}}{2} - \frac{h}{R} (\frac{R^{3}}{3})) d 胃$

$V = {鈭�}_{0}^{2 蟺} (\frac{h R^{2}}{2} - \frac{h R^{2}}{3}) d 胃$

$V = {鈭�}_{0}^{2 蟺} (\frac{h R^{2}}{6}) d 胃$

$= \frac{h R^{2}}{6} (胃)_{0}^{2 蟺}$

$V = \frac{h R^{2}}{6} (2 蟺)$

Hence, required volume is $V = \frac{蟺 R^{2} h}{3}$

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

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.

$鈭� {鈭�}_{R} e^{x y} d A w h e r e R = \{(x, y) I 1 鈮� x 鈮� 2 a n d 1 鈮� y 鈮� 3\}$

Describe the three-dimensional region expressed in each iterated integral:

${鈭�}_{鈭� 2}^{4} 鈥� {鈭�}_{2}^{6} 鈥� {鈭�}_{0}^{5} 鈥� f (x, y, z) d y d x d z$

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

${鈭�}_{- 3}^{3} {鈭�}_{- \sqrt{9 - y^{2}}}^{\sqrt{9 - y^{2}}} {鈭�}_{0}^{9 - y^{2} - z^{2}} f (x, y, z) d x d z d y$

$Express the sum 3 e^{4} + 3 e^{9} + 3 e^{16} + 4 e^{4} + 4 e^{9} + 4 e^{16} using double - summation notation .$

In Exercises 57鈥�60, let R be the rectangular solid defined by

R = {(x, y, z) | 0 鈮� x 鈮� 4, 0 鈮� y 鈮� 3, 0 鈮� z 鈮� 2}.

Assume that the density ofR is uniform throughout.

(a) Without using calculus, explain why the center of mass is (2, 3/2, 1).

(b) Verify that the center of mass is (2, 3/2, 1), using the appropriate integral expressions.

See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.