6. Evaluate Jezdz, where C is the circle |z = 4 taken with counter-clockwise (i.e. anti- clockwise) orientation. Hint: Let C: z(t) = 4 cost+i4 sint for 0≤t≤2. (Ans: 0) 7. Evaluate √ Edz, where C is the circle [2] = 4 taken with counter-clockwise (i.e. anti- clockwise) orientation. (Ans: 32mi) (Ans: 8. Evaluate √(z+1)dz, where C is given by z(t) = cost+isint for 0 ≤t≤π/2. i-2) - 9. Evaluate fezdz, where C is the line segment from i to 1 and 2(t) = + (1 − t)i for (Ans: 1) 10. Evaluate fez²dz, where C is the line segment from 1 to 1+i and z(t) = 1 + it for 0≤t≤1. 11. Evaluate (2-iy²))dz, where C is the upper semicircle C : z(t) 0sts, taken counter-clockwise. 12. Evaluate fc2dz, where C is given by C: z(t)=t+ it² for 0 1), show that and HENCE that (+4) that lim IR = 0. R-+00 dz, where CR is the semicircle below, centre 0, radius R > 2. Show Ri CR -Ri о
6. Evaluate Jezdz, where C is the circle |z = 4 taken with counter-clockwise (i.e. anti- clockwise) orientation. Hint: Let C: z(t) = 4 cost+i4 sint for 0≤t≤2. (Ans: 0) 7. Evaluate √ Edz, where C is the circle [2] = 4 taken with counter-clockwise (i.e. anti- clockwise) orientation. (Ans: 32mi) (Ans: 8. Evaluate √(z+1)dz, where C is given by z(t) = cost+isint for 0 ≤t≤π/2. i-2) - 9. Evaluate fezdz, where C is the line segment from i to 1 and 2(t) = + (1 − t)i for (Ans: 1) 10. Evaluate fez²dz, where C is the line segment from 1 to 1+i and z(t) = 1 + it for 0≤t≤1. 11. Evaluate (2-iy²))dz, where C is the upper semicircle C : z(t) 0sts, taken counter-clockwise. 12. Evaluate fc2dz, where C is given by C: z(t)=t+ it² for 0 1), show that and HENCE that (+4) that lim IR = 0. R-+00 dz, where CR is the semicircle below, centre 0, radius R > 2. Show Ri CR -Ri о
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 92E
Related questions
Question
![6. Evaluate Jezdz, where C is the circle |z = 4 taken with counter-clockwise (i.e. anti-
clockwise) orientation.
Hint: Let C: z(t) = 4 cost+i4 sint for 0≤t≤2.
(Ans: 0)
7. Evaluate √ Edz, where C is the circle [2] = 4 taken with counter-clockwise (i.e. anti-
clockwise) orientation.
(Ans: 32mi)
(Ans:
8. Evaluate √(z+1)dz, where C is given by z(t) = cost+isint for 0 ≤t≤π/2.
i-2)
-
9. Evaluate fezdz, where C is the line segment from i to 1 and 2(t) = + (1 − t)i for
(Ans: 1)
10. Evaluate fez²dz, where C is the line segment from 1 to 1+i and z(t) = 1 + it for
0≤t≤1.
11. Evaluate (2-iy²))dz, where C is the upper semicircle C : z(t)
0sts, taken counter-clockwise.
12. Evaluate fc2dz, where C is given by C: z(t)=t+ it² for 0 <t≤1.
8/15+15/6)
(Ans: -1+21/3)
cost+i sint for
(Ans: 2/3+14/3)
(Ans:
13. Evaluate fez -12dz, where C is the upper half of the circle || = 1 taken with the
counter-clockwise orientation.
(Ans: -4-mi)
14. Evaluate (1/2)dz, where C is the circle |=|=2 taken with clockwise orientation.
Hint: C: z(t)=2 cost-12 sint for 0 St≤2.
(Ans: -2πi)
15. Evaluate c(1/2)dz, where C is the circle |2|= 2 taken with clockwise orientation. (Ans:
0)
16. Evaluate fee'dz, where C is the straight line segment joining 1 to 1+i. (Ans: -2e)
17. Show that cos zdz = sin(1 + i) where C is the polygonal path from 0 to 1+i that
consists of segments from 0 to 1 and 1 to 1+i.
=
18. Show that fee'dze+-1 where C is the straight line segment joining 0 to 1+i.
19. Evaluate √cze dz where C is the square with vertices 0, 1, 1+i and i taken with the
counter-clockwise orientation.
20. If IR=
dz
z²(z+1)'
CR
(a) IR
2
R(R-1)
(b) lim IR = 0.
R-400
21. If IR =
z-5
where CR is the circle |2|= R (R> 1), show that
and HENCE that
(+4)
that lim IR = 0.
R-+00
dz, where CR is the semicircle below, centre 0, radius R > 2. Show
Ri
CR
-Ri
о](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F155158b7-b900-47c4-a397-fbe9c98c96ca%2Fdf4c6e5d-c026-4bf8-b2c9-52649b5ad527%2Fuwna4tl_processed.png&w=3840&q=75)
Transcribed Image Text:6. Evaluate Jezdz, where C is the circle |z = 4 taken with counter-clockwise (i.e. anti-
clockwise) orientation.
Hint: Let C: z(t) = 4 cost+i4 sint for 0≤t≤2.
(Ans: 0)
7. Evaluate √ Edz, where C is the circle [2] = 4 taken with counter-clockwise (i.e. anti-
clockwise) orientation.
(Ans: 32mi)
(Ans:
8. Evaluate √(z+1)dz, where C is given by z(t) = cost+isint for 0 ≤t≤π/2.
i-2)
-
9. Evaluate fezdz, where C is the line segment from i to 1 and 2(t) = + (1 − t)i for
(Ans: 1)
10. Evaluate fez²dz, where C is the line segment from 1 to 1+i and z(t) = 1 + it for
0≤t≤1.
11. Evaluate (2-iy²))dz, where C is the upper semicircle C : z(t)
0sts, taken counter-clockwise.
12. Evaluate fc2dz, where C is given by C: z(t)=t+ it² for 0 <t≤1.
8/15+15/6)
(Ans: -1+21/3)
cost+i sint for
(Ans: 2/3+14/3)
(Ans:
13. Evaluate fez -12dz, where C is the upper half of the circle || = 1 taken with the
counter-clockwise orientation.
(Ans: -4-mi)
14. Evaluate (1/2)dz, where C is the circle |=|=2 taken with clockwise orientation.
Hint: C: z(t)=2 cost-12 sint for 0 St≤2.
(Ans: -2πi)
15. Evaluate c(1/2)dz, where C is the circle |2|= 2 taken with clockwise orientation. (Ans:
0)
16. Evaluate fee'dz, where C is the straight line segment joining 1 to 1+i. (Ans: -2e)
17. Show that cos zdz = sin(1 + i) where C is the polygonal path from 0 to 1+i that
consists of segments from 0 to 1 and 1 to 1+i.
=
18. Show that fee'dze+-1 where C is the straight line segment joining 0 to 1+i.
19. Evaluate √cze dz where C is the square with vertices 0, 1, 1+i and i taken with the
counter-clockwise orientation.
20. If IR=
dz
z²(z+1)'
CR
(a) IR
2
R(R-1)
(b) lim IR = 0.
R-400
21. If IR =
z-5
where CR is the circle |2|= R (R> 1), show that
and HENCE that
(+4)
that lim IR = 0.
R-+00
dz, where CR is the semicircle below, centre 0, radius R > 2. Show
Ri
CR
-Ri
о
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