Chapter 16.4, Problem 8E

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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.

**8.** ∫_{C} y^{4} *dx* + 2*xy*^{3} *dy*,* C* is the ellipse *x*^{2} + 2*y*^{2} = 2

To determine

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

Explanation

**Given data:**

Line integral is
*C* is ellipse

**Formula used:**

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.

The curve *C* is positively oriented, piecewise-smooth, and simply closed curve with domain *D* and hence Greenâ€™s theorem is applicable. The curve ellipse is symmetric about the *x*-axis.

Compare the two expressions

Find the value of

Find the value of

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