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Chapter 14: Q 30. (page 1107)

Evaluate each of the vector field line integrals over the indicated curves.

Short Answer

Expert verified

The required integral is鈭�CF路dr=-8蟺

Step by step solution

01

Given Information

The vector field is F(x,y)=yi-xj.

Cis circle of radius 2centered at origin and traversed clockwise.

02

Solving parametric equations

The parametric equations are

x(t)=2cost,y(t)=2sint,0鈮�t鈮�2蟺

鈬�x'(t)=-2sint,y'(t)=2cost

Hence, C:r(t)=鉄�2cost,2sint鉄�

Also

F(x(t),y(t))=2sinti-2costj

and

r'(t)=x'(t),y'(t)

=鉄�-2sint,2cost鉄�

=(-2sint)i+(2cost)j

03

Finding line integral

The line integral is

鈭�F路dr=鈭�F(x(t),y(t))路r'(t)dt

=鈭�02蟺[2sinti-2costj]路[(-2sint)i+(2cost)j]dt

=鈭�02蟺-4sin2t-4cos2tdt

=鈭�02蟺(-4)dt

=-4[2蟺]

=-8蟺

Hence,鈭�CF路dr=-8蟺

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