Complex Analysis - Maths KU

(–1, –1)
y
C
(2, 8)
x
Figure 5.5 Graph of y = x 3 on the
interval −1 ≤ x ≤ 2
y
y = x 2
x
Figure 5.6 Piecewise smooth path
of integration
5.1 Real Integrals 241
The integrals �
�
G(x, y) ds and C1
manner given in (8) or (11).
C2
G(x, y) ds are then evaluated in the
�Notation
�In
many applications, line integrals appear as a sum
P (x, y) dx + Q(x, y) dy. It is common practice to write this sum as one
C C
integral without parentheses as
�
�
P (x, y) dx + Q(x, y) dy or simply Pdx+ Qdy. (13)
C
A line integral along a closed curve C is usually denoted by
�
Pdx+ Qdy.
C
EXAMPLE 2 C Defined by an Explicit Function
Evaluate �
C xy dx + x2 dy, where C is the graph of y = x 3 , −1 ≤ x ≤ 2.
Solution The curve C is illustrated in Figure 5.5 and is defined by the
explicit function y = x 3 . Hence we can use x as the parameter. Using the
differential dy =3x 2 dx, we apply (9) and (10):
�
C
xy dx + x 2 � 2
dy = x � x 3� dx + x 2 � 3x 2 dx �
−1
� 2
= 4x 4 dx = 4
5 x5
�
�
�
�
−1
2
−1
C
= 132
5 .
EXAMPLE 3 C is a Closed Curve
Evaluate �
xdx, where C is the circle defined by x = cos t, y = sin t,
C
0 ≤ t ≤ 2π.
Solution The differential of x = cos t is dx = − sin tdt, and so from(6),
�
C
� 2π
xdx= cos t (− sin tdt)= 1
2 cos2 �
�
t�
�
0
2π
0
= 1
[1 − 1] = 0.
2
EXAMPLE 4 C is a Closed Curve
Evaluate �
C y2 dx − x 2 dy , where C is the closed curve shown in Figure 5.6.