   Chapter 16.4, Problem 5E

Chapter
Section
Textbook Problem

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.5. ∫C yex dx + 2ex dy, C is the rectangle with vertices (0, 0), (3, 0), (3, 4), and (0, 4)

To determine

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

Explanation

Given data:

Line integral is Cyexdx+2exdy and curve C is a rectangle with vertices (0,0) , (3,0) , (3,4) , and (0,4) .

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={0x30y4 and hence Green’s theorem is applicable.

Compare the two expressions CPdx+Qdy and Cyexdx+2exdy .

P=yexQ=2ex

Find the value of Py .

Py=y(yex)=exy(y)=ex(1){t(t)=1}=ex

Find the value of Qx .

Qx=x(2ex)=2x(ex)=2(ex){t(et)=et}=2ex

Re-modify the equation (1)

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