What is a scalar line integral and how to evaluate it?

 Objective: Evaluate a scalar line integral

Let's consider a simple integral of the form: 

The domain of integration [a b] is a line segment that can be considered as a curve in the x-axis even though it doesn't have a curvature. Let's say that we want to integrate a function over a domain of integration that is not a straight line. The resulting integral is a line integral. A line integral is a type of integral with a domain of integration that can be a line or a curve. The line integral can be the integral of a multivariable function or the the integral of a vector field over a domain of integration that is a curve. In the case of a multivariable function the line integral is called a scalar line integral. If the function is a vector field the line integral is called a vector line integral.

How to evaluate a scalar line integral?   

We divide the curve C into small pieces of curve C₁, C₂, C₃,......Cₙ by the points P₀, P₁, P₂....Pₙ. Each piece has a length Δs₁, Δs₂, Δs₃.....Δsₙ.  Let choose a point P*ᵢ inside the piece Δsᵢ and evaluate f at P we have f(P). Multiplying f(P*i) by Δsᵢ we have the product f(P*i)Δs. Add all the products and let the arc length of the pieces shrink to zero by taking a limit we have:

Calculating the integral using this formula would be cumbersome. The following theorems provide better ways to evaluate a scalar line integral.

Theorem I

Theorem II

Let f be a continuous function with a domain that includes a smooth curve C with parameterization:

Let's substitute: 

in the previous formula. Then we have a definite expression to calculate the line integral:

Example

Solution

Let's use the formula in theorem II : 

Let's calculate f(r,t) and the radical expression:

Let's substitute the given scalar line integral and these 2 expressions in the formula:

Evaluating the integral on the right we have:

Finally the value of the given integral is:

Practice