   Chapter 16, Problem 17RE

Chapter
Section
Textbook Problem

Use Green’s Theorem to evaluate ∫C x2y dx − xy2dy, where C is the circle x2 + y2 = 4 with counterclockwise orientation.

To determine

To Evaluate: Cx2ydxxy2dy using Green’s Theorem.

Explanation

Given data:

Cx2ydxxy2dy .

Here,

C is the circle x2+y2=4 with counterclockwise orientation.

Formula used:

Write the expression Green’s Theorem.

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

Consider the expression as follows.

Cx2ydxxy2dy

Compare left hand side (LHS) of equation (1) with Cx2ydxxy2dy .

P=x2yQ=xy2

Substitute x2y for P and xy2 for Q in equation (1),

Cx2ydxxy2dy=D((xy2)xx2yy)dA (2)

Find the value of limits as follows.

Consider the circle equation as follows.

x2+y2=4 (3)

Write the expression for circle equation.

x2+y2=r2 (4)

Compare equation (3) and (4).

r2=4r=2

Area of circle depends upon radius (r) and angle (θ) . The total angle required to complete one circle is 0 to 2π .

Apply limits and modify equation (2) as follows.

Cx2ydxxy2dy=02π02((xy2)xx2yy)rdrdθ=02π

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