EVALUATION OF LINE INTEGRALS

Fig. 6-2
Fig. 6-2

If the equation of a curve C in the plane  z = 0 is given as  y = f(x), the line integral (14) is evaluated by placing  y = f(x),\ dy = f'(x)dx 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)

which is then evaluated in the usual manner.

Similarly if C is given as x = g(y), then dx = g'(y)dy 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)

If C is given in parametric form x = \phi(t),\ y = \psi(t), 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)

where t_1 and t_2 denote the values of t corresponding to points  A and B respectively.

Combination of the above methods may be used in the evaluation.

Similar methods are used for evaluating line integrals along space curve.