Line Integrals

 

 

In a Nut Shell:  The definite integral you studied in integral calculus

                        b

                       ∫f(x) dx    can be thought of as an integral of f(x) along the x-axis.

                       a

 

Similarly, an integral could be evaluated along a curve in a plane or a curve in space.

Such integrals are called “line integrals”.

 

 

 

Suppose  f(x, y, z)  is a smooth curve in space defined by the parameter t as follows

 

                        x = x(t),   y = y(t),    z = z(t)

 

Now by the Pythagorean theorem    ds  =  √[(dx)2 + (dy)2 + (dz)2]

 

where  ds  represents the differential (arc) length along the curve , s , in space.

 

                            b                          b

So                      ∫ f(x, y, z) ds   =   ∫ f(x, y, z) (ds/dt) dt  =   line integral of  f(x,y,z) along

                           a                           a                                    the curve, s

            

                          b

                         ∫ f(x, y, z) √[(dx/dt)2 + (dy/dt)2 + (dz/dt)2] dt 

                         a

 

 

 

 

 

 

For functions in a plane the vector field is        F(x, y)  =  P(x, y) i  + Q(x, y) j

 

and the vector element along the curve, C, in the plane is     dr  =  dx  i   +  dy  j

 

In this case the dot product

 

            F(x, y)  ·  dr    yields the following line integral along  C in the plane

 

 

              ∫ P(x, y) dx  +    ∫ Q(x, y) dy 

             C                         C                   

 

 

Click here to continue with discussion of line integrals.

 



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