Chapter 16.4, Problem 9E

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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.

**9.** ∫_{C}*y*^{3} *dx* − *x*^{3} *dy*, *C* is the circle *x*^{2} + *y*^{2} = 4

To determine

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

Explanation

**Given data:**

Line integral is
*C* is circle

**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 at origin.

Here,

*r* is radius.

Write the equation of 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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