BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Solutions

Chapter
Section
BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Evaluate the line integral ∫C F · dr, where C is given by the vector function r(t).

21. F(x, y, z) = sin x i + cos y j + xz k,

r(t) = t3it2j + t k, 0 t 1

To determine

To Evaluate: The line integral CFdr .

Explanation

Given data:

The continuous vector field and the vector function are given as follows.

F(x,y,z)=sinxi+cosyj+xzkr(t)=t3it2j+tk,0t1

Formula used:

Write the expression to evaluate the line integral of vector field F(x,y,z) along the curve C .

CFdr=abF(r(t))r(t)dt (1)

Here,

r(t) is the vector function,

a is the lower limit of curve C , and

b is the upper limit of the curve C .

Write the vector function as follows.

r(t)=t3it2j+tk

Write the point (x,y,z) from the vector function as follows.

(x,y,z)=(t3,t2,t)

Write the vector field as follows.

F(x,y,z)=sinxi+cosyj+xzk (2)

Calculation of F(r(t)) :

Substitute t3 for x , t2 for y , t for z in equation (2),

F(x,y,z)=sin(t3)i+cos(t2)j+(t3)(t)k=sin(t3)i+cos(t2)j+(t4)k {cos(θ)=cosθ}

Calculation of r(t) :

To find the derivative of the vector function, differentiate each component of the vector function.

Differentiate each component of the vector function r(t)=t3it2j+tk as follows.

ddt[r(t)]=ddt(t3it2j+tk)

Rewrite the expression as follows.

r(t)=ddt(t3)iddt(t2)j+ddt(t)k=3t2i2tj+k

Calculation of CFdr :

Substitute [sin(t3)i+cos(t2)j+(t4)k] for F(r(t)) , (3t2i2tj+k) for r(t) , 0 for a , and 1 for b in equation (1),

CFdr=01[sin(t3)i+cos(t2)j+(t4)k](3t2i2tj+k)dt=01[sin(t3)i(3t2)i+sin(t3)i(2t)j+sin(t3)ik+cos(t2)j(3t2)i+cos(t2)j(2t)j+cos(t2)jk+(t4)k(3t2)i+(t4)k(2t)j+(t4)kk]dt=01[3t2sin(t3)+0+0+0+(2t)cos(t2)+0+0+0+t4]dt=01(3t2sin(t3)2tcos(t2)+t4)dt

Simplify the expression as follows

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Chapter 16 Solutions

Show all chapter solutions add
Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 P-49ESect-16.2 P-50ESect-16.2 P-51ESect-16.2 P-52ESect-16.3 P-1ESect-16.3 P-2ESect-16.3 P-3ESect-16.3 P-4ESect-16.3 P-5ESect-16.3 P-6ESect-16.3 P-7ESect-16.3 P-8ESect-16.3 P-9ESect-16.3 P-10ESect-16.3 P-11ESect-16.3 P-12ESect-16.3 P-13ESect-16.3 P-14ESect-16.3 P-15ESect-16.3 P-16ESect-16.3 P-17ESect-16.3 P-18ESect-16.3 P-19ESect-16.3 P-20ESect-16.3 P-21ESect-16.3 P-22ESect-16.3 P-23ESect-16.3 P-24ESect-16.3 P-25ESect-16.3 P-26ESect-16.3 P-28ESect-16.3 P-29ESect-16.3 P-30ESect-16.3 P-31ESect-16.3 P-32ESect-16.3 P-33ESect-16.3 P-34ESect-16.3 P-35ESect-16.4 P-1ESect-16.4 P-2ESect-16.4 P-3ESect-16.4 P-4ESect-16.4 P-5ESect-16.4 P-6ESect-16.4 P-7ESect-16.4 P-8ESect-16.4 P-9ESect-16.4 P-10ESect-16.4 P-11ESect-16.4 P-12ESect-16.4 P-13ESect-16.4 P-14ESect-16.4 P-17ESect-16.4 P-18ESect-16.4 P-19ESect-16.4 P-20ESect-16.4 P-21ESect-16.4 P-22ESect-16.4 P-23ESect-16.4 P-24ESect-16.4 P-25ESect-16.4 P-26ESect-16.4 P-27ESect-16.4 P-28ESect-16.4 P-29ESect-16.4 P-30ESect-16.4 P-31ESect-16.5 P-1ESect-16.5 P-2ESect-16.5 P-3ESect-16.5 P-4ESect-16.5 P-5ESect-16.5 P-6ESect-16.5 P-7ESect-16.5 P-8ESect-16.5 P-9ESect-16.5 P-10ESect-16.5 P-11ESect-16.5 P-12ESect-16.5 P-13ESect-16.5 P-14ESect-16.5 P-15ESect-16.5 P-16ESect-16.5 P-17ESect-16.5 P-18ESect-16.5 P-19ESect-16.5 P-20ESect-16.5 P-21ESect-16.5 P-22ESect-16.5 P-23ESect-16.5 P-24ESect-16.5 P-25ESect-16.5 P-26ESect-16.5 P-27ESect-16.5 P-28ESect-16.5 P-29ESect-16.5 P-30ESect-16.5 P-31ESect-16.5 P-32ESect-16.5 P-33ESect-16.5 P-34ESect-16.5 P-35ESect-16.5 P-36ESect-16.5 P-37ESect-16.5 P-38ESect-16.5 P-39ESect-16.6 P-1ESect-16.6 P-2ESect-16.6 P-3ESect-16.6 P-4ESect-16.6 P-5ESect-16.6 P-6ESect-16.6 P-13ESect-16.6 P-14ESect-16.6 P-15ESect-16.6 P-16ESect-16.6 P-17ESect-16.6 P-18ESect-16.6 P-19ESect-16.6 P-20ESect-16.6 P-21ESect-16.6 P-22ESect-16.6 P-23ESect-16.6 P-24ESect-16.6 P-25ESect-16.6 P-26ESect-16.6 P-29ESect-16.6 P-30ESect-16.6 P-33ESect-16.6 P-34ESect-16.6 P-35ESect-16.6 P-36ESect-16.6 P-37ESect-16.6 P-38ESect-16.6 P-39ESect-16.6 P-40ESect-16.6 P-41ESect-16.6 P-42ESect-16.6 P-43ESect-16.6 P-44ESect-16.6 P-45ESect-16.6 P-46ESect-16.6 P-47ESect-16.6 P-48ESect-16.6 P-49ESect-16.6 P-50ESect-16.6 P-51ESect-16.6 P-52ESect-16.6 P-53ESect-16.6 P-54ESect-16.6 P-56ESect-16.6 P-59ESect-16.6 P-60ESect-16.6 P-61ESect-16.6 P-62ESect-16.6 P-63ESect-16.7 P-1ESect-16.7 P-2ESect-16.7 P-3ESect-16.7 P-4ESect-16.7 P-5ESect-16.7 P-6ESect-16.7 P-7ESect-16.7 P-8ESect-16.7 P-9ESect-16.7 P-10ESect-16.7 P-11ESect-16.7 P-12ESect-16.7 P-13ESect-16.7 P-14ESect-16.7 P-15ESect-16.7 P-16ESect-16.7 P-17ESect-16.7 P-18ESect-16.7 P-19ESect-16.7 P-20ESect-16.7 P-21ESect-16.7 P-22ESect-16.7 P-23ESect-16.7 P-24ESect-16.7 P-25ESect-16.7 P-26ESect-16.7 P-28ESect-16.7 P-37ESect-16.7 P-38ESect-16.7 P-39ESect-16.7 P-40ESect-16.7 P-41ESect-16.7 P-42ESect-16.7 P-43ESect-16.7 P-44ESect-16.7 P-45ESect-16.7 P-46ESect-16.7 P-47ESect-16.7 P-48ESect-16.7 P-49ESect-16.8 P-1ESect-16.8 P-2ESect-16.8 P-3ESect-16.8 P-4ESect-16.8 P-5ESect-16.8 P-6ESect-16.8 P-7ESect-16.8 P-8ESect-16.8 P-9ESect-16.8 P-10ESect-16.8 P-11ESect-16.8 P-12ESect-16.8 P-13ESect-16.8 P-14ESect-16.8 P-15ESect-16.8 P-17ESect-16.8 P-18ESect-16.8 P-19ESect-16.8 P-20ESect-16.9 P-1ESect-16.9 P-2ESect-16.9 P-3ESect-16.9 P-4ESect-16.9 P-5ESect-16.9 P-6ESect-16.9 P-7ESect-16.9 P-8ESect-16.9 P-9ESect-16.9 P-10ESect-16.9 P-11ESect-16.9 P-12ESect-16.9 P-13ESect-16.9 P-14ESect-16.9 P-17ESect-16.9 P-18ESect-16.9 P-19ESect-16.9 P-20ESect-16.9 P-23ESect-16.9 P-24ESect-16.9 P-25ESect-16.9 P-26ESect-16.9 P-27ESect-16.9 P-28ESect-16.9 P-29ESect-16.9 P-30ESect-16.9 P-31ESect-16.9 P-32ECh-16 P-1RCCCh-16 P-2RCCCh-16 P-3RCCCh-16 P-4RCCCh-16 P-5RCCCh-16 P-6RCCCh-16 P-7RCCCh-16 P-8RCCCh-16 P-9RCCCh-16 P-10RCCCh-16 P-11RCCCh-16 P-12RCCCh-16 P-13RCCCh-16 P-14RCCCh-16 P-15RCCCh-16 P-16RCCCh-16 P-1RQCh-16 P-2RQCh-16 P-3RQCh-16 P-4RQCh-16 P-5RQCh-16 P-6RQCh-16 P-7RQCh-16 P-8RQCh-16 P-9RQCh-16 P-10RQCh-16 P-11RQCh-16 P-12RQCh-16 P-13RQCh-16 P-1RECh-16 P-2RECh-16 P-3RECh-16 P-4RECh-16 P-5RECh-16 P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

Additional Math Solutions

Find more solutions based on key concepts

Show solutions add

Use the guidelines of this section to sketch the curve. y = ex/x2

Single Variable Calculus: Early Transcendentals, Volume I

Convert the expressions in Exercises 6584 to power form. 18xx23x35

Finite Mathematics and Applied Calculus (MindTap Course List)

Radical Expressions Simplify the expression. 49. (a) 32+18 (b) 75+48

Precalculus: Mathematics for Calculus (Standalone Book)

Evaluate the limit, if it exists. limt01+t1tt

Calculus (MindTap Course List)

Rationalize the denominator: x12x.

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

1. True or false: (a) (b) (c) (d)

Mathematical Applications for the Management, Life, and Social Sciences

An iterated integral for the volume of the solid shown is:

Study Guide for Stewart's Multivariable Calculus, 8th

What rationalizing substitution should be made for x3+2x3+1dx? a) u = x b) u=x3 c) u=x3+2 d) u=x3+1

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

In Exercises 712, refer to the accompanying figure. Which points have negative y coordinates?

Finite Mathematics for the Managerial, Life, and Social Sciences