If C is the line segment that first connects point 2 to point i, and then the spring piece that connects point i to point -1 on the unit circle: Integral=?

Elementary Geometry for College Students
6th Edition
ISBN:9781285195698
Author:Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:Daniel C. Alexander, Geralyn M. Koeberlein
Chapter6: Circles
Section6.3: Line And Segment Relationships In The Circle
Problem 47E
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If C is the line segment that first connects point 2 to point i, and then the spring piece that connects point i to point -1 on the unit circle:

Integral=?

Expert Solution
Step 1

Consider two regions C1 and Cthen these can be defined as follows,

Advanced Math homework question answer, step 1, image 1

Thus, C=C1UC2

Advanced Math homework question answer, step 1, image 2

The conjugate of C1=z(t) is (2-2t)-it and the conjugate of C2 is e-it.

Differentiate the C1 with respect to t.

Advanced Math homework question answer, step 1, image 3

First, obtain the value of the integral of Cand substitute the value of dz.

Advanced Math homework question answer, step 1, image 4

Differentiate the C2 with respect to t.

Advanced Math homework question answer, step 1, image 5

Obtain the value of the integral of C2 and substitute the value of dz.

Advanced Math homework question answer, step 1, image 6

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