If the equation of a curve C in the plane is given as , the line integral (14) is evaluated by placing in the integrand to obtain the definite integral
which is then evaluated in the usual manner.
Similarly if C is given as , then and the line integral becomes
If C is given in parametric form , the line integral becomes
where and denote the values of corresponding to points and respectively.
Combination of the above methods may be used in the evaluation.
Similar methods are used for evaluating line integrals along space curve.
is given as
, the line integral (14) is evaluated by placing
in the integrand to obtain the definite integral
![\displaystyle \int_{a_1}^{a_2}[P\{x, f(x)\}dx + Q\{x, f(x)\}f'(x)dx] \cdots(19)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7Ba_1%7D%5E%7Ba_2%7D%5BP%5C%7Bx%2C+f%28x%29%5C%7Ddx+%2B+Q%5C%7Bx%2C+f%28x%29%5C%7Df%27%28x%29dx%5D+%5Ccdots%2819%29&bg=T&fg=000000&s=0 "\displaystyle \int_{a_1}^{a_2}[P\{x, f(x)\}dx + Q\{x, f(x)\}f'(x)dx] \cdots(19)")
, then
and the line integral becomes
![\displaystyle \int_{b_1}^{b_2}[P\{g(y), y\}g'(y)dy + Q\{g(y), y\}dy]\cdots(20)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7Bb_1%7D%5E%7Bb_2%7D%5BP%5C%7Bg%28y%29%2C+y%5C%7Dg%27%28y%29dy+%2B+Q%5C%7Bg%28y%29%2C+y%5C%7Ddy%5D%5Ccdots%2820%29&bg=T&fg=000000&s=0 "\displaystyle \int_{b_1}^{b_2}[P\{g(y), y\}g'(y)dy + Q\{g(y), y\}dy]\cdots(20)")
, the line integral becomes
![\displaystyle \int_{t_1}^{t_2} [P\{ \phi(t),\ \psi(t) \}\phi'(t)dt + Q\{ \phi(t),\ \psi(t) \}\psi'(t)dt] \cdots (21)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+%5BP%5C%7B+%5Cphi%28t%29%2C%5C+%5Cpsi%28t%29+%5C%7D%5Cphi%27%28t%29dt+%2B+Q%5C%7B+%5Cphi%28t%29%2C%5C+%5Cpsi%28t%29+%5C%7D%5Cpsi%27%28t%29dt%5D+%5Ccdots+%2821%29&bg=T&fg=000000&s=0 "\displaystyle \int_{t_1}^{t_2} [P\{ \phi(t),\ \psi(t) \}\phi'(t)dt + Q\{ \phi(t),\ \psi(t) \}\psi'(t)dt] \cdots (21)")
and
denote the values of
corresponding to points
and
respectively.
