   Chapter 16.4, Problem 8E

Chapter
Section
Textbook Problem

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.8. ∫C y4 dx + 2xy3 dy, C is the ellipse x2 + 2y2 = 2

To determine

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

Explanation

Given data:

Line integral is Cy4dx+2xy3dy and curve C is ellipse x2+2y2=2 .

Formula used:

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

CPdx+Qdy=D(QxPy)dA (1)

Here,

Py is continuous first-order partial derivative of P,

Qx is continuous first-order partial derivative of 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 Green’s theorem is applicable. The curve ellipse is symmetric about the x-axis.

Compare the two expressions CPdx+Qdy and Cy4dx+2xy3dy .

P=y4Q=2xy3

Find the value of Py .

Py=y(y4)=4y3 {t(tn)=ntn1}

Find the value of Qx

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