Question

Find the value of the line integral (Hint: if F is conservative, the integration may be easier on an alternative path.)

Answer 1

Step 1:

The function is .

Definition of the line integral of a vector field :

If F is a continuous vector field on a smooth curve C, the function in the interval .

Then the line integral of F on C  is .

(a)

The function is and in the interval .

Then

Then the line integral of F on C  is

Step 2:

(b)

The function is and in the interval .

Then

Then the line integral of F on C  is

Step 3:

(c)

The function is and in the interval .

Then

Then the line integral of F on C  is

Solution :

(a) The line integral is 0.

(b) The line integral is .

(c) The line integral is .