   Chapter 16.2, Problem 19E

Chapter
Section
Textbook Problem

# Evaluate the line integral ∫C F · dr, where C is given by the vector function r(t).19. F(x, y) = xy2 i − x2 j,r(t) = t3i + t2j, 0 ⩽ t ⩽ 1

To determine

To Evaluate: The line integral CFdr when the continuous vector field F(x,y)=(xy2)i(x2)j and the vector function r(t)=(t3)i+(t2)j,0t1 .

Explanation

Given data:

Consider the given continuous vector field and the vector function.

F(x,y)=(xy2)i(x2)jr(t)=(t3)i+(t2)j,0t1

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)=(t3)i+(t2)j

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

(x,y)=(t3,t2)

Write the vector field as follows.

F(x,y)=(xy2)i(x2)j (2)

Calculation of F(r(t)) :

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

F(r(t))=(t3)(t2)2i(t3)2j=(t3)(t4)i(t6)j=(t7)i(t6)j

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)=(t3)i+(t2)j as follows.

ddt[r(t)]=ddt[(t3)i+(t2)j]

Rewrite the expression as follows

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