Scalar line integral: Independence of parameterization

 Objective: Show that the value of a line integral doesn't change if the parameterization of the curve changes. 

In the previous post we calculated the value of a line integral. What if the parameterization of the curve changes? Let's use the same example and change the parameterization of the curve?

Example

  Solution  

Let's apply the formula to evaluate a line integral:

Let's calculate f(r(t)) by substituting x, y and z in the function:

Let's calculate the radical expression:

Let's substitute f(x,y.z), f(r(t)) and the radical expression in the formula:

We obtain the same result as in the previous example. Scalar line integrals are independent of parameterization. 

Practice