   Chapter 16.2, Problem 23E

Chapter
Section
Textbook Problem

Use a calculator to evaluate the line integral correct to four decimal places.23. ∫C F · dr, where F ( x ,   y ) = x + y   i + ( y / x ) j and r(t) = sin2 t i + sin t cos t j, π/6 ⩽ t ⩽ π/3

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)=x+yi+(yx)jr(t)=sin2ti+sintcostj,π6tπ3

Formula used:

Write the expression to evaluate the line integral of vector field F(x,y) 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)=sin2ti+sintcostj

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

(x,y)=(sin2t,sintcost)

Write the vector field as follows.

F(x,y)=x+yi+(yx)j (2)

Calculation of F(r(t)) :

Substitute sin2t for x and sintcost for y in equation (2),

F(x,y)=sin2t+sintcosti+(sintcostsin2t)j=sin2t+sintcosti+(costsint)j=sin2t+sintcosti+cottj

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)=sin2ti+sintcostj as follows.

ddt[r(t)]=ddt(sin2ti+sintcostj)

Rewrite the expression as follows

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