Q 42 The iterated integrals use cylin... [FREE SOLUTION]

Chapter 13: Q 42 (page 1066)

The iterated integrals use cylindrical coordinates. Describe the solids determined by the limits of integration.
${鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{1} {鈭�}_{0}^{\sqrt{1 - r^{2}}} f (r, 胃, z) r d z d r d 胃$

Short Answer

Expert verified

The region represents inside the circle with equation $r = 1$ bounded aabove by the sphere with radius 1 centered at the origin in the $r - 胃$ plane

Step by step solution

Step 1:Given information

The given expression is ${鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{1} {鈭�}_{0}^{\sqrt{1 - r^{2}}} f (r, 胃, z) r d z d r d 胃$

Step 2:Simplification

Given

${鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{1} {鈭�}_{0}^{\sqrt{1 - r^{2}}} f (r, 胃, z) r d z d r d 胃$

From the limits of z

$z = \sqrt{1 - r^{2}}$

$z^{2} = 1 - r^{2}$

$z^{2} + r^{2} = 1$

$x^{2} + y^{2} + z^{2} = 1$

This equation represents the equation of sphere centered at origin

From the limits of r

$r = 1$

$r^{2} = 1$

$x^{2} + y^{2} = 1$

This equation represents the equation of circle with radius 1

The limit $胃$ varies from $0 t o \frac{蟺}{2}$

Hence, the region represents inside the circle with equation $r = 1$ bounded aabove by the sphere

with radius 1 centered at the origin in the $r - 胃$ plane

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

The iterated integrals use cylindrical coordinates. Describe the solids determined by the limits of integration.
${鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{1} {鈭�}_{0}^{\sqrt{1 - r^{2}}} f (r, 胃, z) r d z d r d 胃$

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Calculus

Geometry

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

91影视

The iterated integrals use cylindrical coordinates. Describe the solids determined by the limits of integration.鈭�0蟺2鈭�01鈭�01-r2fr,胃,zrdzdrd胃

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Calculus

Geometry

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

The iterated integrals use cylindrical coordinates. Describe the solids determined by the limits of integration.
${鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{1} {鈭�}_{0}^{\sqrt{1 - r^{2}}} f (r, 胃, z) r d z d r d 胃$