   Chapter 16.4, Problem 30E

Chapter
Section
Textbook Problem

Complete the proof of the special case of Green’s Theorem by proving Equation 3.

To determine

To prove: The special case of Green’s theorem CQdy=DQxdA .

Explanation

Given data:

Consider the expression for special case of Green’s theorem as,

CQdy=DQxdA

Consider a type II region as D={(x,y)|f1(y)xf2(y),cyd} and functions f1 and f2 are continuous.

The type II region is shown in Figure 1.

From Figure 1, the curves are,

C1Qdy=dcQ(f1(y),y)dyC2Qdy=0C3Qdy=cdQ(f2(y),y)dyC4Qdy=0

Find the value of DQxdA .

DQxdA=cdf1(y)f2(y)Qxdxdy=cd[Q(x)]f1(y)f2(y)dy{Atdt=A(t)}

DQxdA=cd[Q(f2(y),y)Q(f1(y),y)]dy (1)

Find the value of CQdy

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