Chapter 16.2, Problem 21E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### 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) = t3i − t2j + 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

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