/*! 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 33. Evaluate the integral:聽鈭琒F(x,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Q 33. (page 1151)

Evaluate the integral:

鈭�SF(x,y,z)路nds

F(x,y,z)=sinycoszi+yz2j+zx2kand S is the surface of the region W bounded by the paraboloidy=x2+z2and the planesy=1andy=4

Short Answer

Expert verified

The integral is鈭�SF(x,y,z)路ndS=21蟺2

Step by step solution

01

Given Information

It is given that F(x,y,z)=sinycoszi+yz2j+zx2k

We need to evaluate鈭�SF(x,y,z)路ndS

02

Use of Divergence Theorem to evaluate the Integral

Theorem states that:

Let W be a bounded region in R3whose boundary S is a smooth or piecewise-smooth closed oriented surface. If a vector field role="math" F(x,y,z)is defined on an open region containing W, then,

鈭�SF(x,y,z)路ndS=鈭�WdivF(x,y,z)dV

n is outwards unit normal vector.

03

Finding the Divergence

The vector field is F(x,y,z)=sinycoszi+yz2j+zx2k

It is written as:

F(x,y,z)=F1(x,y,z)i+F2(x,y,z)j+F3(x,y,z)k

Finding the divergence

divF(x,y,z)=鈭�鈭�xi+鈭�鈭�yj+鈭�鈭�zk路F1i+F2j+F3k

=鈭�F1鈭�x+鈭�F2鈭�y+鈭�F3鈭�z

Hence, the divergence is

divF(x,y,z)=鈭�鈭�xi+鈭�鈭�yj+鈭�鈭�zk路sinycoszi+yz2j+zx2k

=鈭�鈭�x(sinycosz)+鈭�鈭�yyz2+鈭�鈭�zzx2

=0+z2+x2

=x2+z2

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

Compute dS for your parametrization in Exercise 9.

ComputethedivergenceofthevectorfieldsinExercises17鈥�22.G(x,y)=xcos(xy),ycos(xy)

Integrate the given function over the accompanying surface in Exercises 27鈥�34.f(x,y,z)=ez, where S is the portion of the unit sphere in the first octant.

In what way is Green鈥檚 Theorem a special case of Stokes鈥� Theorem?

Generalize your answer to Exercise 12 to give a parametrization and a normal vector for the extension of any differentiable plane curve y = f(x) through a 鈮� z 鈮� b.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.