   Chapter 16.4, Problem 9E

Chapter
Section
Textbook Problem

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.9. ∫C y3 dx − x3 dy, C is the circle x2 + y2 = 4

To determine

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

Explanation

Given data:

Line integral is Cy3dxx3dy and curve C is circle x2+y2=4 .

Formula used:

Green’s Theorem:

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.

Write the expression for circle with center at origin.

x2+y2=r2

Here,

Write the equation of circle.

x2+y2=4x2+y2=22

Compare the equations x2+y2=r2 and x2+y2=22 .

r=2

Hence, consider circle parametric equations as x=r , y=0 , and dA=rdrdθ .

The curve C is positively oriented, piecewise-smooth, and simply closed curve with domain D={0r20θ2π and hence Green’s theorem is applicable.

Compare the two expressions CPdx+Qdy and Cy3dxx3dy .

P=y3Q=x3

Find the value of Py .

Py=y(y3)=3y2 {t(tn)=ntn1}

Find the value of Qx

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