Example 1 :
If determine the value of where C is the curve in the xy plane from (0, 0) to (1, 2).
The curve lies in xy plane, so, z = 0. z can never be taken as independent variable z is a dependent variable. Now, out of x and y, any one variable can be taken as independent.
Suppose x is taken as independent variable
So, the line integral reduces to a definite integral.
If y is taken as independent variable then x can be expressed in terms of y as
So, the line integral reduces to a definite integral
Example : 1
Evaluate around a circle
Let C denotes the circle. The parametric equations of circle is
Here, x and y have been expressed in terms of parameter which varies from 0 to as one traverses the circle.
Here, r is a constant, because integral is carried over a circle.