/*! 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 74. Let R聽be the radius of a sphere... [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 74. (page 1068)

Let $R$ be the radius of a sphere. Use cylindrical coordinates to set up and evaluate a triple integral proving that the volume of the sphere is $\frac{4}{3} {蟺搁}^{3}$ .

Short Answer

Expert verified

Volume can be evaluated using the relation between cylindrical and cartesian coordinates.

Step by step solution

01

Given Information

It is given that radius of sphere is $R$ .

02

Evaluation of limits

We know that

$r = \sqrt{x^{2} + y^{2}}, \tan 胃 = \frac{y}{x}, z = z$

and

$x = r \cos 胃, y = r \sin 胃, z = z$

Cylindrical limits are $- R 鈮� r 鈮� R, 0 鈮� 胃鈮� 2 蟺, - \sqrt{R^{2} - r^{2}} 鈮� z 鈮� \sqrt{R^{2} - r^{2}}$

03

Calculation of Volume

The volume of sphere is

$V = {鈭�}_{V} r d r d 胃 d z$

$= {鈭�}_{r = - R}^{R} {鈭�}_{胃 = 0}^{2 蟺} {鈭�}_{z = - \sqrt{R^{2} - r^{2}}}^{\sqrt{R^{2} - r^{2}}} r d r d 胃 d z$

$= {鈭�}_{r = - R}^{R} {鈭�}_{胃 = 0}^{2 蟺} 2 \sqrt{R^{2} - r^{2}} r d r d 胃$

$= \{{鈭�}_{胃 = 0}^{2 蟺} d 胃\} 脳 \{(- 1) {鈭�}_{r = 0}^{R} \sqrt{R^{2} - r^{2}} (- 2 r) d r\}$

Apply the limits

$V = (2 蟺) 脳 {\{(- 1) \frac{{(R^{2} - r^{2})}^{3 / 2}}{3 / 2}\}|}_{r = 0}^{R}$

$V = (\frac{4 蟺}{3}) (0 - (- R^{3}))$

$V = (\frac{4 蟺 R^{3}}{3})$

Hence proved.

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

Use the results of Exercises 59 and 60 to find the centers of masses of the lamin忙 in Exercises 61鈥�67.

Use the lamina from Exercise 61, but assume that the density is proportional to the distance from the x-axis.

Explain how to construct a Riemann sum for a function of three variables over a rectangular solid.

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

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

Find the masses of the solids described in Exercises 53鈥�56.

The solid bounded above by the paraboloid with equation $z = 8 - x^{2} - y^{2}$ and bounded below by the rectangle $R = {(x, y, 0) | 1 鈮� x 鈮� 2 and 0 鈮� y 鈮� 2}$ in the xy-plane if the density at each point is proportional to the square of the distance of the point from the origin.

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

$鈭� {鈭�}_{R} (x - e^{y}) d A W h e r e R = {(x, y) | - 3 鈮� x 鈮� 2, - 2 鈮� y 鈮� 2}$

See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

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.