Complex Analysis - Maths KU

244 Chapter 5 Integration in the Complex Plane
In Problems 15–18, evaluate �
to (2, 5).
C
(2x + y) dx + xy dy on the given curve from (−1, 2)
15. y = x +3 16. y = x 2 +1
17.
y
18.
(2, 5)
(–1, 2)
(2, 2)
x
(–1, 2)
(–1, 0)
Figure 5.9 Figure for Problem 17
Figure 5.10 Figure for Problem 18
�
In Problems 19–22, evaluate ydx+ xdyon the given curve from (0, 0) to (1, 1).
C
19. y = x 2
20. y = x
y
(2, 5)
x
(2, 0)
21. C consists of the line segments from (0, 0) to (0, 1) and from (0, 1) to (1, 1).
22. C consists of the line segments from (0, 0) to (1, 0) and from (1, 0) to (1, 1).
�
� 2 2
23. Evaluate 6x +2y � dx +4xy dy, where C is given by x = √ t, y = t,
4 ≤ t ≤ 9.
�
24. Evaluate
�
25. Evaluate
C
C
to (1, 1).
�
26. Evaluate
to (9, 2).
C
C
�
In Problems 27 and 28, evaluate
−y 2 dx + xy dy, where C is given by x =2t, y = t 3 , 0 ≤ t ≤ 2.
2x 3 ydx+(3x + y) dy, where C is given by x = y 2 from (1, −1)
4xdx +2ydy, where C is given by x = y 3 + 1 from (0, −1)
27. 28.
y
x 2 + y 2 = 4
Figure 5.11 Figure for Problem 27
x
C
� x 2 + y 2 � dx − 2xy dy on the given closed curve.
y
y = √x
(1, 1)
y = x 2
Figure 5.12 Figure for Problem 28
x