   Chapter 16.2, Problem 22E

Chapter
Section
Textbook Problem

Evaluate the line integral ∫C F · dr, where C is given by the vector function r(t).22. F(x, y, z) = x i + y j + xy k,r(t) = cos t i + sin t j + t k, 0 ⩽ t ⩽ π

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)=xi+yj+xykr(t)=costi+sintj+tk,0tπ

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)=costi+sintj+tk

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

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

Write the vector field as follows.

F(x,y,z)=xi+yj+xyk (2)

Calculation of F(r(t)) :

Substitute cost for x and sint for y in equation (2),

F(x,y,z)=costi+sintj+(cost)(sint)k=costi+sintj+costsintk

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)=costi+sintj+tk as follows.

ddt[r(t)]=ddt(costi+sintj+tk)

Rewrite the expression as follows

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