Page 240 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 10] LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 231

3. If the integrand is a perfect differential, then it may be evaluated through knowledge of the end
points (that is, without reference to any particular joining curve). (See the section on indepen-
dence of path on Page 232; also see Page 237.)

These techniques are further illustrated below for plane curves and for three space in the problems.

EVALUATION OF LINE INTEGRALS FOR PLANE CURVES
If the equation of a curve C in the plane z ¼ 0is given as y ¼ f ðxÞ, the line integral (2)is evaluated
by placing y ¼ f ðxÞ; dy ¼ f ðxÞ dx in the integrand to obtain the definite integral
0
ð
a 2
Pfx; f ðxÞg dx þ Qfx; f ðxÞg f ðxÞ dx ð7Þ
0
a 1
which is then evaluated in the usual manner.
Similarly, if C is given as x ¼ gðyÞ, then dx ¼ g ðyÞ dy and the line integral becomes
0
ð
b 2
Pfgð yÞ; ygg ð yÞ dy þ Qfgð yÞ; yg dy ð8Þ
0
b 1
If C is given in parametric form x ¼ ðtÞ; y ¼ ðtÞ, the line integral becomes
ð
t 2
Pf ðtÞ; ðtÞg ðtÞ dt þ Qf ðtÞ; ðtÞg; ðtÞ dt ð9Þ
0
0
t 1
where t 1 and t 2 denote the values of t corresponding to points A and B, respectively.
Combinations of the above methods may be used in the evaluation. If the integrand A dr is a
perfect differential, d , then
ð ð
ðc;dÞ
A dr ¼ d ¼ ðc; dÞ ða; bÞ ð6Þ
C ða;bÞ
Similar methods are used for evaluating line integrals along space curves.

PROPERTIES OF LINE INTEGRALS EXPRESSED FOR PLANE CURVES
Line integrals have properties which are analogous to those of ordinary integrals. For example:

ð ð ð
1: Pðx; yÞ dx þ Qðx; yÞ dy ¼ Pðx; yÞ dx þ Qðx; yÞ dy
C C C
ð ð
ða 2 ;b 2 Þ ða 1 ;b 1 Þ
2: Pdx þ Qdy ¼ Pdx þ qdy
ða 1 ;b 1 Þ ða 2 ;b 2 Þ
Thus, reversal of the path of integration changes the sign of the line integral.

ð ð ð
ða 2 ;b 2 Þ ða 3 ;b 3 Þ ða 2 ;b 2 Þ
3: Pdx þ Qdy ¼ Pdx þ Qdy þ Pdx þ Qdy
ða 2 ;b 1 Þ ða 1 ;b 1 Þ ða 3 ;b 3 Þ
where ða 3 ; b 3 Þ is another point on C.

Similar properties hold for line integrals in space.