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Chapter 14: Q 31. (page 1107)

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

F(x,y)and Care as in Exercise 29, but Cis traversed in the reverse direction, fromx=10tox=5

Short Answer

Expert verified

The required integral is鈭�CF路dr=70.

Step by step solution

01

Given Information

The given vector field is F(x,y)=i-j

The curve is parametrized as

y-x2=1for10鈮�x鈮�5

02

Solving parametrized equations

The parametric equations of curve is:

x(t)=t,y(t)=1+t2,10鈮�t鈮�5

role="math" localid="1653593890857" 鈬�x'(t)=1,y'(t)=2t

Hence,

C:r(t)=t,1+t2

Given is F(x(t),y(t))=i-j

and

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

=鉄�1,2t鉄�

=(1)i+(2t)j

03

Finding line integral

The formula is

鈭�CF路dr=鈭�CF(x(t),y(t))路r'(t)dt

=鈭�105[(1)i+(-1)j]路[(1)i+(2t)j]dt

=鈭�105(1-2t)dt

=t-t2105

=[5-25]-[10-100]

=-20+90

=70

Therefore,鈭�CF路dr=70

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