The line integrals can be evaluated in two ways, depending on whether the curve is defined parametrically or by an explicit function. In either case the basic idea is to convert the line integral to a definite integral in a single variable.
Method of Evaluation – Curve Defined Parametrically
If is a smooth curve parameterized by
, then we simply replace
and
in the integral by the functions
and
, and the appropriate differential
or
by
, or
. The expression
is called the differential of arc length. The integration is carried out with respect to the variable
in the usual manner:
(1)
(2)
(3)
Method of Evaluation – Curve Defined by an Explicit Function
If the curve is defined by an explicit function
we can use
as a parameter. With
and
, the foregoing line integrals become, in turn,
(4)
(5)
(6)
YouTube Videos
Watch these two awesome videos on line integral and its physical interpretation.
Reference
Dennis G. Zill. Advanced Engineering Mathematics, edition. Jones
Bartlett Learning. 2016.