Question

Show that the value of is the same for each parametric representation of C.

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

. 

Solution :

(a) .

(b) .