Complex Analysis - Maths KU

5.2 Complex Integrals 245
In Problems 29 and 30, evaluate �
29. 30.
y
(–1, 1)
(–1, –1)
(1, 1)
(1, –1)
x
C x2 y 3 dx − xy 2 dy on the given closed curve.
Figure 5.13 Figure for Problem 29
Figure 5.14 Figure for Problem 30
31. Evaluate � � 2 2
x − y C
� ds, where C is given by x = 5 cos t, y = 5 sin t, 0≤ t ≤ 2π.
32. Evaluate �
ydx−xdy, where C is given by x = 2 cos t, y = 3 sin t, 0 ≤ t ≤ π.
−C
33. Verify that the line integral �
C y2 dx + xy dy has the same value on C for each
of the following parametrizations:
y
(2, 4)
C : x =2t +1, y =4t +2, 0 ≤ t ≤ 1
C : x = t 2 , y =2t 2 , 1 ≤ t ≤ √ 3
C : x =lnt, y =2lnt, e ≤ t ≤ e 3 .
34. Consider the three curves between (0, 0) and (2, 4):
Show that �
C1
C : x = t, y =2t, 0 ≤ t ≤ 2
C : x = t, y = t 2 , 0 ≤ t ≤ 2
C : x =2t− 4, y =4t− 8, 2 ≤ t ≤ 3.
�
� �
xy ds = xy ds, but xy ds �= xy ds. Explain.
C3
35. If ρ(x, y) is the density of a wire (mass per unit length), then the mass of
the wire is m = �
ρ(x, y) ds. Find the mass of a wire having the shape of
C
a semicircle x = 1 + cos t, y = sin t, 0 ≤ t ≤ π, if the density at a point P is
directly proportional to the distance from the y-axis.
36. The coordinates of the center of mass of a wire with variable density are given
by ¯x = My/m, ¯y = Mx/m where
�
�
�
m = ρ(x, y) ds, Mx = yρ(x, y) ds, My = xρ(x, y) ds.
5.2 Complex Integrals
C
Find the center of mass of the wire in Problem 35.
In the preceding 5.2 section we reviewed two types of real integrals. We saw that the definition
of the definite integral starts with a real function y = f(x) that is defined on an interval on
the x-axis. Because a planar curve is the two-dimensional analogue of an interval, we then
generalized the definition of � b
f(x) dx to integrals of real functions of two variables defined
a
on a curve C in the Cartesian plane. We shall see in this section that a complex integral is
defined in a manner that is quite similar to that of a line integral in the Cartesian plane.
Since curves play a big part in the definition of a complex integral, we begin with a
brief review of how curves are represented in the complex plane.
C
C1
C2
x
C