   Chapter 16.4, Problem 6E

Chapter
Section
Textbook Problem

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.6. ∫C (x2 + y2) dx + (x2 − y2) dy,C is the triangle with vertices (0, 0), (2, 1), and (0, 1)

To determine

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

Explanation

Given data:

Line integral is C(x2+y2)dx+(x2y2)dy and curve C is a triangle with vertices (0,0) , (2,1) , and (0,1) .

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 a continuous first-order partial derivative of P,

Qx is a continuous first-order partial derivative of Q, and

P and Q have continuous partial derivatives.

Draw the curve as shown in Figure 1.

From Figure 1, the curve C is positively oriented, piecewise-smooth, and simply closed curve with domain D={0x2y0y1 and hence Green’s theorem is applicable.

Compare the two expressions CPdx+Qdy and C(x2+y2)dx+(x2y2)dy .

P=x2+y2Q=x2y2

Find the value of Py .

Py=y(x2+y2)=y(x2)+y(y2)=0+2y{t(k)=0,t(t2)=2t}=2y

Find the value of Qx .

Qx=x(x2y2)=x(x2)+x(y2)=2x0{t(k)=0,t(t2)=2t}=2x

Re-modify the equation (1).

CPdx+Qdy=x1x2y1y2(QxPy)dydx

Substitute x2+y2 for P, x2y2 for Q, 2y for Py , 2x for Qx , 0 for x1 , 2y for x2 , 0 for y1 , and 1 for y2

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