   Chapter 16.2, Problem 24E

Chapter
Section
Textbook Problem

Use a calculator to evaluate the line integral correct to four decimal places.24. ∫C F · dr, where F(x, y, z) = yzex i + zxey j + xyez k and r(t) = sin t i + cos t j + tan t k, 0 ⩽ t ⩽ π/4

To determine

To Evaluate: The line integral CFdr when the continuous vector field F(x,y,z)=yzexi+zxeyj+xyezk and the vector function r(t)=sinti+costj+tantk,0tπ4 .

Explanation

Given data:

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

F(x,y,z)=yzexi+zxeyj+xyezkr(t)=sinti+costj+tantk,0tπ4

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)=sinti+costj+tantk

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

(x,y,z)=(sint,cost,tant)

Write the vector field as follows.

F(x,y,z)=yzexi+zxeyj+xyezk (2)

Calculation of F(r(t)) :

Substitute sint for x , cost for y , tant for z in equation (2) as follows.

F(x,y,z)=costtantesinti+tantsintecostj+sintcostetantk=cost(sintcost)esinti+tantsintecostj+sintcostetantk=sintesinti+tantsintecostj+sintcostetantk

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)=sinti+costj+tantk as follows.

ddt[r(t)]=ddt(sinti+costj+tantk)

Rewrite the expression as follows.

r(t)=ddt(sint)i+ddt(cost)j+ddt(tant)k=costi+(sint)j+sec2tk=costisintj+sec2tk

Calculation of CFdr :

Substitute (sintesinti+tantsintecostj+sintcostetantk) for F(r(t)) , (costisintj+sec2tk) for r(t) , 0 for a , and π4 for b in equation (1)

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