Chapter 16.4, Problem 10E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.

**10.** ∫* _{C}* (1 −

To determine

**To evaluate**: The line integral using Green’s Theorem.

Explanation

**Given data:**

Line integral is
*C* is a boundary between circles

**Formula used:**

**Greenâ€™s Theorem:**

Consider a positively oriented curve *C* which is piece-wise smooth, simple closed curve in plane with domain *D*. Then,

Here,

*P*,

*Q*, and

*P* and *Q* have continuous partial derivatives.

Write the expression for circle with center origin.

Here,

*r* is radius.

Write the equation of first circle.

Compare the equations

Write the equation of second circle.

Compare the equations

Hence, consider circle parametric equations as

The curve *C* is positively oriented, piecewise-smooth, and simply closed curve with domain

Compare the two expressions

Find the value of

Find the value of

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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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